Mechanical movement – change in the position of a body in space over time relative to other bodies.
Forward movement – movement in which all points of the body follow the same trajectories.
Material point – a body whose dimensions can be neglected under given conditions, since its dimensions are negligible compared to the distances under consideration.
Trajectory – line of body movement.(Trajectory equation – dependence y(x))
Path l(m) – trajectory length.Properties: l ≥ 0 , does not decrease!
Moving s(m) – a vector connecting the initial and final position of the body.
https://pandia.ru/text/78/241/images/image003_70.gif" width="141" height="33"> sX= x – x0- movement module
Properties: s ≤ l, s = 0 in a closed area. l
Speed u(m/s)– 1) average path u =; average displacement = ; ;
2) instantaneous - the speed at a given point can only be found using the speed equation uX = u0x + aXt or according to schedule u(t)
Acceleration a(m/s2) - change in speed per unit time.
https://pandia.ru/text/78/241/images/image009_44.gif" width="89" height="52 src=">.gif" width="12" height="23 src="> - accelerated linear motion
() If ↓ - slow motion straight
If ^ - circular movement
Relativity of motion - dependence on choice reference systems: trajectories, displacements, speeds, acceleration of mechanical movement.
Galileo's principle of relativity - all laws of mechanics are equally valid in all inertial systems countdown.
The transition from one reference system to another is carried out according to the rule:
https://pandia.ru/text/78/241/images/image019_30.gif" width="32" height="33 src=">.gif" width="19" height="32 src=">. gif" width="20" height="32">
Where u1 - the speed of the body relative to a fixed frame of reference,
u2 – speed of the moving reference frame,
urel (υ12) – speed of the 1st body relative to the 2nd.
Types of movement.
Straight-line movement .
Rectilinear uniform motion. | Rectilinear uniformly accelerated motion. |
|
accelerated slow |
|
against the axis |
accelerated slow |
sx= uxt | sx=u0xt+ or sx = without t! |
against the Ox axis | ux=uox+a xt ux along the Ox axis ux slow motion by oh accelerated accelerated against the Ox axis |
a = 0 Oh |
fast motion slow motion |
x =x0 +uxt x axis
x =x0 +u0xt+ x x
ux =const ux along the Ox axis
a
x =constAhahCurvilinear movement .
Circular motion with a constant modulus speed | Parabolic motion with acceleration |
2πRn(m/s) - linear speed 2πn(rad/s) – angular velocity i.e. u = ω R
T = – period (s), T =
| x = xo + uoxt +; y = yo + uoyt + ux= uox+ gxt ; uy= uoy+ gyt uоx = u0 cosa uоy = u0 sina
|
(m/s2) - centripetal acceleration
n= – frequency (Hz=1/s), n =
ySpecial cases uniformly accelerated motion under the influence of gravity .
Vertical movement. | Movement of a body thrown horizontally. |
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1. If u0 = 0 ; u= gt 2. If u0, the body moves upward ; u= u 0-gt If u0, the body falls down from a height ; u= - u 0 + GT 3. If u0 ↓ ; u= u 0+gt (Oy axis is directed downwards) |
Training tasks.1(A) In what case can a projectile be taken as a material point: a) calculation of projectile flight range; b) calculation of the shape of the projectile, ensuring a reduction in air resistance. 1) Only in the first case. 2) Only in the second case. 3) In both cases. 4) Neither in the first nor in the second case. 2(A) A wheel rolls down a flat hill in a straight line. What trajectory describes the center of the wheel relative to the road surface? 1) Circle. 3) Spiral. 2) Cycloid. 4) Direct. 3(A) What is the displacement of a point moving in a circle of radius R when it is rotated by 90º? 1) R/2 2) R 3) 2R 4) R 4(A) Which of the graphs can be a graph of the distance traveled by the body?
https://pandia.ru/text/78/241/images/image104_5.gif" width="12 height=152" height="152"> https://pandia.ru/text/78/241/images/image109_6.gif"> A 7(A) The car travels at speed half the time u 1, and the second half of the time at speed u 2, moving in the same direction. What is the average speed of the car? 8(A) The equation for the dependence of the coordinates of a moving body on time has the form: X = 4 - 5t + 3t2 (m). What is the equation for the projection of the velocity of a body? 1) u x = - 5 + 6t (m/s) 3) u x = - 5t + 3t2 (m/s) 2) u x = 4 - 5t (m/s) 4) ux = - 5t + 3t (m/s) 9(A) The parachutist descends vertically downward at a constant speed u =7 m/s. When he is at a height h = 160 m, a lighter falls out of his pocket. The time it takes for the lighter to fall to the ground is 1) 4 s 2) 5 s 3) 8 ss 10(A) If a body that begins to move uniformly accelerated from a state of rest covers the distance S in the first second, then in the fourth second it covers the distance 1) 3S 2) 5S 3) 7S 4) 9S 11(A) At what speed do two cars move away from each other when driving away from an intersection along mutually perpendicular roads at speeds of 40 km/h and 30 km/h? 1) 50km/h 2) 70km/hkm/hkm/h 12(A) Two objects move according to the equations u x1 = 5 - 6t (m/s) and x2 = 1 - 2t + 3t2 (m). Find the magnitude of their velocity relative to each other 3 s after the start of movement. 1) 3 m/cm/cm/s 4) 6 m/s 13(A) When accelerating from a state of rest, the car acquired a speed of 12 m/s, having traveled 36 m. If the acceleration of the car is constant, then 5 s after the start its speed will be equal to 1) 6 m/s 2) 8 m/cm/cm/s 14(A) Two skiers start with an interval ∆t. The speed of the first skier is 1.4 m/s, the speed of the second skier is 2.2 m/s. If the second skier catches up with the first in 1 minute, then the interval ∆t is equal to 1) 0.15 min 3) 0.8 min 2) 0.6 min 4) 2.4 min 15(A) A ball is thrown with an initial speed of 30 m/s. The entire flight time of the ball at the throwing angle α=45º is equal to 1) 1.2 s 2) 2.1 s 3) 3.0 s 4) 4.3 s 16(A) A stone is thrown from a tower with an initial speed of 8 m/s in the horizontal direction. Its speed will become equal to 10 m/s later 1) 0.6 s 2) 0.7 s 3) 0.8 s 4) 0.9 s 17(B) A material point moves at a constant speed along a circle of radius R. How will the physical quantities listed in the first column change if the rotation frequency of the point decreases? acceleration 3) will not change B) Circulation period circumferentially 18(B) Two material points move in circles of radii R1 and R2 with R2 = 4 R1. If the linear velocities of the points are equal, their ratio centripetal accelerations a1/a2 equals …… 19(B) Using the graph of the body's speed as a function of time, determine the average speed for the entire period of movement. Indicate the accuracy of the result to the nearest tenth.
20(C) Inclined plane intersects the horizontal plane along straight line AB. The angle between the planes is α=30º. A small washer begins to move up an inclined plane from point A with an initial speed u0 = 2 m/s at an angle β=60º to straight line AB. Find the maximum distance that the puck moves away from straight AB while climbing the inclined plane. Neglect the friction between the washer and the inclined plane.
Answers to training tasks.Test tasks.1 (A) A material point is: 1) a body of negligible mass; 2) the body is very small; 3) a point showing the position of the body in space; 4) a body whose dimensions can be neglected in the conditions of this problem. 2(A) What is the change in position of one body relative to another called: 1) trajectory; 2) moving; 4) mechanical movement. 3(A) What is the displacement of a point moving in a circle of radius R when it rotates 180º? 1) 5 mm 3) 12.5 mm 8(A) The equation for the dependence of the projection of the displacement of a moving body on time has the form: sx = 10t + 4t2 (m). What is the equation for the coordinates of a body that started moving from a point with coordinate 5? 1) x = 5+10t+2t2 (m) 3) x = 5+10t+4t2 (m) 2) x = 5+5t+2t2 (m) 4) x = 5+10t+2t2 (m) 9(A) A crane lifts a load vertically upward with a certain speed u0. When the load is at a height h = 24 m, the crane cable breaks and the load falls to the ground in 3 s. At what speed will the weight fall to the ground? 1) 32 m/cm/cm/s 4) 21.5 m/s 10(A) A body that begins to move uniformly accelerated from a state of rest with an acceleration of 2 m/s2, then in the third second it will cover the distance 1) 7 m 2) 5 m 3) 3 m 4) 2 m https://pandia.ru/text/78/241/images/image139_2.gif" width="12" height="120">1) 40 m/s x, m 12(A) The escalator staircase rises up with a speed u, at what speed relative to the walls should a person go down it in order to rest relative to the people standing on the stairs going down? 1) u 2) 2u 3) 3u 4) 4u 13(A) At a speed of 12 m/s, the braking time of a truck is 4 s. If, when braking, the acceleration of the car is constant and does not depend on the initial speed, then when braking, the car will reduce its speed from 18 m/s to 15 m/s, having passed 1) 12.3 m 3) 28.4 m 2) 16.5 m 4) 33.4 m 14(A) Along the roundabout highway 5 km long, a truck and a motorcyclist are traveling in one direction at speeds u1, respectively = 40 km/h and u2 = 100 km/h. If in starting moment time they were in the same place, then the motorcyclist will catch up with the car after passing 1) 3.3 km 3) 8.3 km 2) 6.2 km 4) 12.5 km 15(A) A body was thrown from the Earth's surface at an angle α to the horizon with an initial speed u0 = 10 m/s, if the body’s flight range is L = 10 m, then the angle α is equal to 1) 15º 2) 22.5º 3) 30º 4) 45º 16(A) A boy threw a ball horizontally from a window located at a height of 20 m. The ball fell at a distance of 8 m from the wall of the house. At what initial speed was the ball thrown? 1) 0.4 m/s 2) 2.5 m/s 3) 3 m/s 4) 4 m/s 17(B) A material point moves at a constant speed along a circle of radius R. How will the physical quantities listed in the first column change if the speed of the point increases? Physical quantities. Their change. A) Angular velocity 1) will increase B) Centripetal 2) will decrease acceleration 3) will not change B) Circulation period circumferentially |
,
If t1 = t2 = … = tn u1 u2
numerically equal to the displacement or distance traveled.
5. Wheel movement without slipping.
5(A)
The figure shows a bus schedule from point A to point B and back. Point A is at point X= 0, and point B is at point X= 30 km. What is the maximum ground speed of the bus along the entire route there and back?
1) 2.4 m/s2 uх, m/s
υ, m/s
Test tasks.
1 (A) A material point is:
1) a body of negligible mass;
2) the body is very small;
3) a point showing the position of the body in space;
4) a body whose dimensions can be neglected in the conditions of this problem.
2(A) What is the change in position of one body relative to another called:
1) trajectory;
2) moving;
4) mechanical movement.
3(A) What is the displacement of a point moving in a circle of radius R when it rotates 180º?
1) R/2 2) R 3) 2R 4) R
4(A) The line that a body describes when moving in space is called:
1) trajectory;
2) moving;
4) mechanical movement.
5(A)
The figure shows a graph of the movement of a body from point A to point B and back. Point A is located at point x 0 = 30 m, and point B is at point x = 5 m. What is the minimum speed of the bus along the entire route there and back?
1) 5.2 m/s Hm
6(A) The body begins to decelerate in a straight line with uniform acceleration along the Ox axis. Indicate the correct location of the velocity and acceleration vectors at time t.
7(A) Located on horizontal surface The speed of the table is 5 m/s. Under the action of friction, the block moves with an acceleration equal in magnitude to 1 m/s 2 . What is the distance traveled by the block in 6 s?
1) 5 m 2) 12 m 3) 12.5 m 4) 30 m
8(A) The equation for the dependence of the projection of the displacement of a moving body on time has the form: s x = 10t + 4t 2 (m).What is the equation for the coordinates of a body that began moving from a point with coordinate 5?
1) x = 5+10t+2t 2 (m) 3) x = 5+10t+4t 2 (m)
2) x = 5+5t+2t 2 (m) 4) x = 5+5t+4t 2 (m)
9(A) A crane lifts a load vertically upward at a certain speed u 0 . When the load is at a height h = 24 m, the crane cable breaks and the load falls to the ground in 3 s. At what speed will the weight fall to the ground?
1) 32 m/s 2) 23 m/s 3) 20 m/s 4) 21.5 m/s
10(A) A body that begins to move uniformly accelerated from a state of rest with an acceleration of 2 m/s 2, then in the third second it will cover the distance
1) 7 m 2) 5 m 3) 3 m 4) 2 m
11(A) The coordinates of bodies A and B moving along the same straight line change over time, as shown in the graph. What is the speed of body A relative to body B?
1) 40 m/s x, m
12(A) The escalator staircase rises up with a speed u, at what speed relative to the walls should a person go down it in order to rest relative to the people standing on the stairs going down?
1) u 2) 2u 3) 3u 4) 4u
13(A) At a speed of 12 m/s, the braking time of a truck is 4 s. If, when braking, the acceleration of the car is constant and does not depend on the initial speed, then when braking, the car will reduce its speed from 18 m/s to 15 m/s, having passed
1) 12.3 m 3) 28.4 m
2) 16.5 m 4) 33.4 m
14(A) A truck and a motorcyclist are traveling along a 5 km long ring road in one direction at speeds u 1, respectively. = 40 km/h u 2 = 100 km/h. If at the initial moment of time they were in the same place, then the motorcyclist will catch up with the car, passing
1) 3.3 km 3) 8.3 km
2) 6.2 km 4) 12.5 km
15(A) A body was thrown from the Earth's surface at an angle α to the horizon with an initial speed u 0 = 10 m/s, if the body’s flight range is L = 10 m, then the angle α is equal to
1) 15º 2) 22.5º 3) 30º 4) 45º
16(A) A boy threw a ball horizontally from a window located at a height of 20 m. The ball fell at a distance of 8 m from the wall of the house. At what initial speed was the ball thrown?
1) 0.4 m/s 2) 2.5 m/s 3) 3 m/s 4) 4 m/s
17(B) A material point moves at a constant speed along a circle of radius R. How will the physical quantities listed in the first column change if the speed of the point increases?
Physical quantities. Their change.
In this lesson, the topic of which is “Determining the coordinates of a moving body,” we will talk about how you can determine the location of a body and its coordinates. Let's talk about reference systems, consider an example problem, and also remember what movement is
Imagine: you threw a ball with all your might. How to determine where he will be in two seconds? You can wait two seconds and just see where he is. But, even without looking, you can approximately predict where the ball will be: the throw was stronger than usual, directed at a large angle to the horizon, which means it will fly high, but not far... Using the laws of physics, it will be possible to accurately determine the position of our ball.
Determining the position of a moving body at any time is the main task of kinematics.
Let's start with the fact that we have a body: how to determine its position, how to explain to someone where it is? We will say about a car: it is on the road 150 meters before the traffic light or 100 meters after the intersection (see Fig. 1).
Rice. 1. Determining the location of the machine
Or on the highway 30 km south of Moscow. Let's say about the phone on the table: it is 30 centimeters to the right of the keyboard or next to the far corner of the table (see Fig. 2).
Rice. 2. Position the phone on the table
Note: we will not be able to determine the position of the car without mentioning other objects, without being attached to them: a traffic light, a city, a keyboard. We define position, or coordinates, always relative to something.
Coordinates are a set of data from which the position of an object and its address are determined.
Examples of ordered and unordered names
The coordinate of the body is its address at which we can find it. It's orderly. For example, knowing the row and the place, we determine exactly where our place is in the cinema hall (see Fig. 3).
Rice. 3. Cinema hall
A letter and a number, for example e2, precisely defines the position of the piece on the chessboard (see Fig. 4).
Rice. 4. Position of the piece on the board
Knowing the address of the house, for example, Solnechnaya Street 14, we will look for it on this street, on even side, between houses 12 and 16 (see Fig. 5).
Rice. 5. Searching for a home
The street names are not ordered; we will not search for Solnechnaya Street alphabetically between Rozovaya and Turgenev streets. Also, telephone numbers and car license plates are not organized (see Fig. 6).
Rice. 6. Unordered names
These consecutive numbers are just a coincidence and do not mean proximity.
We can set the body position in different systems coordinates, as convenient for us. For the same car, you can set the exact geographical coordinates(latitude and longitude) (see Fig. 7).
Rice. 7. Longitude and latitude of the area
Rice. 8. Location relative to a point
Moreover, if we select different such points, we will get different coordinates, although they will specify the position of the same car.
So, the position of the body is relative different bodies will be different in different coordinate systems. What is movement? Movement is a change in body position over time. Therefore, we will describe movement in different reference systems in different ways, and there is no point in considering the movement of a body without a reference system.
For example, how does a glass of tea move on a table on a train if the train itself is moving? It depends on what. Relative to the table or the passenger sitting next to him on the seat, the glass is at rest (see Fig. 9).
Rice. 9. Movement of the glass relative to the passenger
Regarding the tree about railway the glass moves with the train (see Fig. 10).
Rice. 10. Movement of the glass along with the train relative to the tree
Relative to the earth's axis, the glass and the train along with all the points earth's surface will also move in a circle (see Fig. 11).
Rice. 11. Movement of the glass with the rotation of the Earth relative to the Earth’s axis
Therefore, there is no point in talking about movement in general; movement is considered in relation to the reference system.
Everything we know about the movement of a body can be divided into observable and calculable. Let's remember the example of the ball that we threw. The observable is its position in the chosen coordinate system when we first throw it (see Fig. 12).
Rice. 12. Observation
This is the moment in time when we abandoned him; time that has passed since the throw. Even if there is no speedometer on the ball that would show the speed of the ball, its module, as well as its direction, can also be found out using, for example, slow motion.
Using observed data, we can predict, for example, that a ball will fall 20 m from where it was thrown after 5 seconds or hit the top of a tree after 3 seconds. The position of the ball at any given time is, in our case, calculated data.
What determines each new position of a moving body? It is defined by displacement, because displacement is a vector that characterizes a change in position. If the beginning of the vector is combined with the initial position of the body, then the end of the vector will point to the new position of the moved body (see Fig. 13).
Rice. 13. Motion vector
Let's look at several examples of determining the coordinates of a moving body based on its movement.
Let the body move rectilinearly from point 1 to point 2. Let's construct a displacement vector and designate it (see Fig. 14).
Rice. 14. Body movement
The body moved along one straight line, which means that one coordinate axis directed along the movement of the body will be enough for us. Let's say we are observing the movement from the side, let's align the origin with the observer.
Displacement is a vector; it is more convenient to work with projections of vectors on the coordinate axes (we have one). - vector projection (see Fig. 15).
Rice. 15. Vector projection
How to determine the coordinate of the starting point, point 1? We lower the perpendicular from point 1 to the coordinate axis. This perpendicular will intersect the axis and mark the coordinate of point 1 on the axis. We also determine the coordinate of point 2 (see Fig. 16).
Rice. 16. Lower perpendiculars to the OX axis
The displacement projection is equal to:
With this direction of the axis and the displacement will be equal in magnitude to the displacement itself.
Knowing the initial coordinate and displacement, finding the final coordinate of the body is a matter of mathematics:
The equation
An equation is an equality containing an unknown term. What is its meaning?
Any problem is that we know something, but we don’t know something, and the unknown needs to be found. For example, a body from a certain point moved 6 m in the direction of the coordinate axis and ended up at a point with coordinate 9 (see Fig. 17).
Rice. 17. Initial position of the point
How to find from what point the body began to move?
We have a pattern: the displacement projection is the difference between the final and initial coordinates:
The meaning of the equation will be that we know the displacement and the final coordinate () and can substitute these values, but we do not know the initial coordinate, it will be unknown in this equation:
And already solving the equation, we will get the answer: initial coordinate.
Let's consider another case: the movement is directed to the side, opposite direction coordinate axes.
Coordinates of the initial and end points are determined in the same way as before - perpendiculars are lowered onto the axis (see Fig. 18).
Rice. 18. The axis is directed in the other direction
The displacement projection (nothing changes) is equal to:
Note that is greater than , and the displacement projection when directed against the coordinate axis will be negative.
The final coordinate of the body from the equation for the displacement projection is equal to:
As we can see, nothing changes: in the projection onto the coordinate axis, the final position is equal to the initial position plus the displacement projection. Depending on which direction the body has moved, the projection of the movement will be positive or negative in a given coordinate system.
Let's consider the case when the displacement and the coordinate axis are directed at an angle to each other. Now one coordinate axis is not enough for us; we need a second axis (see Fig. 19).
Rice. 19. The axis is directed in the other direction
Now the displacement will have a non-zero projection on each coordinate axis. These displacement projections will be defined as before:
Note that the module of each of the projections in this case is less than the displacement module. We can easily find the displacement module using the Pythagorean theorem. It can be seen that if you build right triangle(see Fig. 20), then its legs will be equal to and , and the hypotenuse is equal to the displacement module or, as is often written, simply .
Rice. 20. Pythagorean triangle
Then, using the Pythagorean theorem, we write:
The car is located 4 km east of the garage. Use one coordinate axis pointing east, with the origin at the garage. Enter the coordinates of the car in given system in 3 minutes, if the car was traveling at a speed of 0.5 km/min to the west during this time.
The problem does not say anything about the car turning or changing speed, so we consider the motion to be uniform and rectilinear.
Let's draw a coordinate system: the origin is at the garage, the x axis is directed to the east (see Fig. 21).
The car was initially at the point and was moving west according to the conditions of the problem (see Fig. 22).
Rice. 22. Car movement to the west
The displacement projection, as we have repeatedly written, is equal to:
We know that the car traveled 0.5 km every minute, which means that to find the total movement, we need to multiply the speed by the number of minutes:
This is where physics ends, all that remains is to express it mathematically the desired coordinate. Let's express it from the first equation:
Let's substitute the displacement:
All that remains is to plug in the numbers and get the answer. Don't forget that the car was moving west against the x-axis direction, which means that the velocity projection is negative: .
The problem is solved.
The main thing we used today to determine the coordinate is the expression for the displacement projection:
And from it we have already expressed the coordinate:
In this case, the displacement projection itself can be specified, can be calculated as , as in the problem of uniform rectilinear motion, it can be calculated more complexly, which we still have to study, but in any case, the coordinate of the moving body (where the body ended up) can be determined from the initial coordinate (where the body was) and according to the projection of movement (where it moved).
This concludes our lesson, goodbye!
Bibliography
- Sokolovich Yu.A., Bogdanova G.S. Physics: A reference book with examples of problem solving. - 2nd edition, revision. - X.: Vesta: Ranok Publishing House, 2005. - 464 p.
- Peryshkin A.V., Gutnik E.M. Physics: 9th grade. Tutorial for educational institutions. - 14th ed. - M.: Bustard, 2009.
- Class-fizika.narod.ru ().
- Av-physics.narod.ru ().
- Class-fizika.narod.ru ().
Homework
- What is movement, path, trajectory?
- How can you determine the coordinates of a body?
- Write down the formula to determine the displacement projection.
- How will the displacement module be determined if the displacement has projections on two coordinate axes?
Topic No. 1. Kinematics.
Mechanical movement – change in the position of a body in space over time relative to other bodies.
Forward movement –movement in which all points of the body follow the same trajectories.
Material point – a body whose dimensions can be neglected under given conditions, because its dimensions are negligible compared to the distances under consideration.
Trajectory – line of body movement.(Trajectory equation – dependence y(x))
Path l (m) – trajectory length.Properties: l ≥ 0, does not decrease!
Moving s(m) – a vector connecting the initial and final position of the body.
s x = x – x 0- projection length of the displacement vector
Properties: s≤ l, s = 0 in a closed area. l
Speed u (m/s)– 1) average path u = ; average displacement = ; ;
2) instantaneous - the speed at a given point can only be found using the speed equation u x = u 0x + a x t or according to schedule u(t)
Acceleration a(m/s 2) - change in speed per unit time.
; = if - motion accelerated rectilinear
( )If ↓ - slow motion straight
If ^ - circular movement
Relativity of motion- dependence on the choice of reference system: trajectory, displacement, speed, acceleration of mechanical movement.
Galileo's principle of relativity– all laws of mechanics are equally valid in all inertial frames of reference.
The transition from one reference system to another is carried out according to the rule:
And = -
Where u 1 - the speed of the body relative to a fixed frame of reference,
u 2 – speed of the moving frame of reference,
u rel (υ 12) – speed of the 1st body relative to the 2nd.
Types of movement.
Straight-line movement.
| Rectilinear uniform motion. | Rectilinear uniformly accelerated motion. |
| x o =const x s x | x o x x o x s x s x accelerated slow |
 x = x 0 + u x t x along the x axis ~ t x 0 t against the axis |  x = x 0 + u 0 x t + x x x ~ t 2 x o x o t t accelerated slow
|
| s x = u x t | s x =u 0 x t + or s x = without t! |
 u x = const u x along the Ox axis t against the Ox axis | u x = u ox + a x t u x along the Ox axis u x u o u o slow motion by oh υ = 0 t t accelerated accelerated against the Ox axis |
| a = 0 a x t | a x = const Ahah t t |
Curvilinear movement.
| Circular motion with a constant modulus speed | Parabolic motion with free fall acceleration. |
=2πRn(m/s) - linear speed =2πn(rad/s) – angular speed i.e. u = ω R
 (m/s 2) - centripetal acceleration T = – period (s), T =  n= – frequency (Hz=1/s), n =
| x = x o + u ox t + ; y = y o + u oy t + u x = u ox + g x t ; u y = u oy + g y t u o x = u 0 cosa u o y = u 0 sina g x = 0 g y = - g  y u x u y s x |
Special cases of uniformly accelerated motion under the influence of gravity.
Additional Information
for special cases of problem solving.
1. Decomposition of a vector into projection.  The magnitude of the vector can be found using the Pythagorean theorem: S = | 2. Average speed. 1) by definition  2) for 2 x S; if 3)  ,
  if t 1 = t 2 = … = t n u 1 u 2 |
3. Area method. On the chart u x (t) area of the figure  numerically equal to the displacement or distance traveled. S = S 1 - S 2 ℓ = S 1 + S 2 | 4. Physical meaning of the derivative. For coordinate equations x(t) And y(t) → u x = x΄, u y = y΄, and A x = u΄ x = x΄΄, A y = u΄ y = y΄΄, |
 5. Wheel movement without slipping. u post = u rotation
(if there is no slippage)  The speed of a point on the rim of a wheel relative to the ground. | 6. Flight range. Flight range is maximum at a throwing angle of 45˚ υ 0 = const |
S 1: S 2: S 3: …: S n = 1: 3: 5: 7: ….: (2n-1)
S n = S 1 (2n – 1) = (2n - 1)
 |
S 1: S 2: S 3: …: S n = 1 2: 2 2: 3 2: 4 2: ….: n 2
S n = S 1 n 2 = n 2
Training tasks.
1(A) Two problems are solved:
a) the docking maneuver of two spacecraft is calculated;
b) the period of revolution of spacecraft around the Earth is calculated.
In what case can spaceships be considered as material points?
1) Only in the first case.
2) Only in the second case.
3) In both cases.
4) Neither in the first nor in the second case.
2(A) The wheel rolls down a flat hill in a straight line. What trajectory does a point on the wheel rim describe relative to the road surface?
1) Circle. 3) Spiral.
2) Cycloid. 4) Direct.
3(A) What is the displacement of a point moving in a circle of radius R when it is rotated by 60º?
1) R/2 2) R 3) 2R 4) R
Note: draw a drawing, mark two positions of the body, the movement will be a chord, analyze how the triangle will turn out (all angles are 60º).
4(A) How far will the boat travel when making a complete turn with a radius of 2 m?
1) 2 m 3) 6.28 m
2) 4 m 4) 12.56 m
Note: make a drawing, the path here is the length of the semicircle.
5(A)
The figure shows a bus schedule from point A to point B and back. Point A is at point X= 0, and point B is at point X= 30 km. What is the maximum ground speed of the bus along the entire route there and back?
6(A) The body begins to move rectilinearly with uniform acceleration along the Ox axis. Indicate the correct location of the velocity and acceleration vectors at time t.
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Note: with rectilinear motion, vectors v and a are directed along one straight line, with increasing speed they are co-directed.
7(A) The car travels half the distance at speed u 1, and the second half of the journey at a speed u 2 ,
Note: This problem is a special case of finding the average speed. The derivation of the formula comes from the definition
, where s 1 = s 2, and t 1 = and t 2 =
8(A) The equation for the dependence of the projection of the speed of a moving body on time has the form: u x = 3-2t(m/s).What is the projection equation for the displacement of a body?
1) s x =2t 2 (m) 3) s x =2t-3t 2 (m)
2) s x =3t-2t 2 (m) 4) s x =3t-t 2 (m)
Note: write down the equation for the speed of uniformly accelerated motion in general form and, comparing it with the data in the problem, find what u 0 and a are equal to, insert these data into the equation of displacement written in general form.
9(A) How far will a body freely falling from rest travel in the fifth second? Take the free fall acceleration to be 10 m/s 2 .
1) 45 m 2) 55 m 3) 125 m 4) 250 m
Note: write down the expression h for the case u o =0, the desired h= h 5 - h 4, where h for 5 s and 4 s, respectively.
10(A) If a body that begins to move uniformly accelerated from a state of rest covers the distance S in the first second, then in the first three seconds it covers the distance
1) 3S 2) 4S 3) 8S 4) 9S
Note: use the motion properties of uniformly accelerated motion for u 0 =0
11(A) Two cars are moving towards each other at speeds of 20 m/s and 90 km/h, respectively. What is the absolute speed of the first relative to the second?
1) 110 m/s 2) 60 m/s 3) 45 m/s 4) 5 m/s
Note: Relative speed is the difference between vectors, because the velocity vectors are directed oppositely, it is equal to the sum of their modules.
12(A) An observer from the shore sees that a swimmer is crossing a river with a width of h = 189 m perpendicular to the shore. In this case, the speed of the river is u=1.2 m/s, and the speed of the swimmer relative to the water is u=1.5 m/s. The swimmer will cross the river for...
1) 70 s 2) 98 s 3) 126 s 4) 210 s
Note: construct a velocity triangle based on = + , go to the Pythagorean theorem, express from it the speed of the swimmer relative to the shore, and find the time with it.
13(A) At a speed of 10 m/s, the braking time of a truck is 3 s. If, when braking, the acceleration of the car is constant and does not depend on the initial speed, then when braking, the car will reduce its speed from 16 m/s to 9 m/s in ...
1) 1.5 s 2) 2.1 s 3) 3.5 s 4) 4.5 s
Note: from considering the first situation, find the acceleration and substitute it into the velocity equation for the second situation, from which you can express the required time.
14(A) A motor ship departs from the pier, moving at a constant speed of 18 km/h; after 40 s, a boat departs from the same pier in pursuit with an acceleration of 0.5 m/s 2 . How long will it take for it to catch up with the ship, moving with constant acceleration?
1) 20 s 2) 30 s 3) 40 s 4) 50 s
Note: take the time of movement of the boat as t, then the time of movement of the motor ship is t+40, write down the expressions for the displacement of the motor ship (uniform motion) and the boat (uniformly accelerated motion) and equate them. Solve the resulting square quadratic equation relative to t. Don't forget to convert the units 18 km/h = 5 m/s.
15(A) Two people play a ball, throwing it at an angle α=60º to the horizontal. The ball is in flight t = 2 s. In this case, the distance at which the players are located is equal to
1) 9.5 m 2) 10 m 3) 10.5 m 4) 11.5 m
Note: make a drawing - in x,y axes– the trajectory is a parabola, the point of intersection of the parabola with the x-axis corresponds to the flight range, at this point the equation x(t) has the form s=u o cos60º t. To find u 0, use the equation y(t), which at the same point has the form 0=u o sin60º t- . From this equation, express u o and substitute it into the first equation. The calculation formula looks like
16(A) The plane flies with cargo to its destination at an altitude of 405 m above sandy terrain with a horizontal profile at a speed of 130 m/s. In order for the cargo to reach the intended place on the ground (neglect the force of movement resistance), the pilot must release it from the fasteners before reaching the target
1) 0.53 km 3) 0.95 km
2) 0.81 km 4) 1.17 km
Note: Consider in theory the example “Motion of a body thrown horizontally.” From the expression for flight altitude, express the time of fall and substitute it into the flight range formula.
17(B) A material point moves at a constant speed along a circle of radius R, making one revolution in time T. How will the physical quantities listed in the first column change if the radius of the circle increases and the period of revolution remains the same?
Physical quantities. Their change.
A) Speed 1) will increase
B) Angular velocity 2) will decrease
C) Centripetal 3) will not change
acceleration
| A | B | IN |
Note: write down the defining formulas of the proposed quantities in terms of R and analyze their mathematical dependence, taking into account the constancy of the period. The numbers in the right column can be repeated.
18(B) What is the linear speed of a point on the surface of the globe corresponding to 60º north latitude? The radius of the Earth is 6400 km. Give the answer in m/s, round to whole numbers.
Note: make a drawing and note that the point at the indicated latitude rotates relative to the earth's axis in a circle with radius r = R earth cos60º.
19(B) υ, m/s
Note: The simplest way to find a path through the area of a figure under a graph. A complex figure can be represented as the sum of two trapezoids and one rectangle.
20(C) = 2 m/s at an angle β=60º to straight line AB. During its motion, the puck moves onto straight line AB at point B. Neglecting friction between the puck and the inclined plane, find the distance AB.
Note: to solve the problem, you should consider the trajectory of the puck - a parabola lying on an inclined plane and select the coordinate axes, see Fig.
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In t.B x=s and the equation x(t) has the form s=u o cos60º t
You can find t from the equation у(t), at this point it will look like 0=u o sin60ºt – . By solving this system of equations together, find s.
Answers to training tasks.
| 1A | 2A | 3A | 4A | 5A | 6A | 7A | 8A | 9A | 10A |
| 11A | 12A | 13A | 14A | 15A | 16A | 17V | 18V | 19V | 20C |
| 69 cm |
Training tasks.
1(A) In what case can a projectile be taken as a material point:
a) calculation of projectile flight range;
b) calculation of the shape of the projectile, ensuring a reduction in air resistance.
1) Only in the first case. 2) Only in the second case.
3) In both cases. 4) Neither in the first nor in the second case.
2(A) The wheel rolls down a flat hill in a straight line. What trajectory
describes the center of the wheel relative to the road surface?
1) Circle. 3) Spiral.
2) Cycloid. 4) Direct.
3(A) What is the displacement of a point moving in a circle of radius R when it is rotated by 90º?
1) R/2 2) R 3) 2R 4) R
4(A) Which of the graphs can be a graph of the distance traveled by the body?
5(A) The figure shows a graph of the projection of the speed of movement of the body. What is the absolute value of the minimum acceleration of the body along the entire path?
1) 2.4 m/s 2 u x, m/s
6(A) A body moves uniformly in a circle. Specify the correct location of the vectors linear speed and acceleration in t.A.
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7(A) The car travels at speed half the time u 1, and the second half of the time at a speed u 2 , moving in the same direction. What is the average speed of the car?
8(A) The equation for the dependence of the coordinates of a moving body on time has the form:
X = 4 - 5t + 3t 2 (m).What is the equation for the projection of the body's velocity?
1) u x = - 5 + 6t (m/s) 3) u x = - 5t + 3t 2 (m/s)
2) u x = 4 - 5t (m/s) 4) u x = - 5t + 3t (m/s)
9(A) The parachutist descends vertically downward at a constant speed u =7 m/s. When he is at a height h = 160 m, a lighter falls out of his pocket. The time it takes for the lighter to fall to the ground is
1) 4 s 2) 5 s 3) 8 s 4) 10 s
10(A) If a body that begins to move uniformly accelerated from a state of rest covers the distance S in the first second, then in the fourth second it covers the distance
1) 3S 2) 5S 3) 7S 4) 9S
11(A) At what speed do two cars move away from each other when driving away from an intersection along mutually perpendicular roads at speeds of 40 km/h and 30 km/h?
1) 50 km/h 2) 70 km/h 3) 10 km/h 4) 15 km/h
12(A) Two objects move according to the equations u x 1 = 5 - 6t (m/s) and x 2 = 1 - 2t + 3t 2 (m). Find the magnitude of their velocity relative to each other 3 s after the start of movement.
1) 3 m/s 2) 29 m/s 3) 20 m/s 4) 6 m/s
13(A) When accelerating from a state of rest, the car acquired a speed of 12 m/s, having traveled 36 m. If the acceleration of the car is constant, then 5 s after the start its speed will be equal to
1) 6 m/s 2) 8 m/s 3) 10 m/s 4) 15 m/s
14(A) Two skiers start with an interval ∆t. The speed of the first skier is 1.4 m/s, the speed of the second skier is 2.2 m/s. If the second skier catches up with the first in 1 minute, then the interval ∆t is equal to
1) 0.15 min 3) 0.8 min
2) 0.6 min 4) 2.4 min
15(A) A ball is thrown with an initial speed of 30 m/s. The entire flight time of the ball at the throwing angle α=45º is equal to
1) 1.2 s 2) 2.1 s 3) 3.0 s 4) 4.3 s
16(A) A stone is thrown from a tower with an initial speed of 8 m/s in the horizontal direction. Its speed will become equal to 10 m/s later
1) 0.6 s 2) 0.7 s 3) 0.8 s 4) 0.9 s
17(B) A material point moves at a constant speed along a circle of radius R. How will the physical quantities listed in the first column change if the rotation frequency of the point decreases?
acceleration 3) will not change
B) Circulation period
circumferentially
| A | B | IN |
18(B) Two material points move in circles with radii R 1 and R 2 and R 2 = 4 R 1 . If the linear velocities of the points are equal, the ratio of their centripetal accelerations a 1 /a 2 equals ……
19(B) Using the graph of the body's speed as a function of time, determine the average speed for the entire period of movement. Indicate the accuracy of the result to the nearest tenth.
υ, m/s
20(C) An inclined plane intersects a horizontal plane along straight line AB. The angle between the planes is α=30º. A small washer begins to move up an inclined plane from point A with an initial speed u 0 = 2 m/s at an angle β=60º to straight line AB. Find the maximum distance that the puck moves away from straight AB while climbing the inclined plane. Neglect the friction between the washer and the inclined plane.
Answers to training tasks.
| 1A | 2A | 3A | 4A | 5A | 6A | 7A | 8A | 9A | 10A |
| 11A | 12A | 13A | 14A | 15A | 16A | 17V | 18V | 19V | 20C |
| 21.7 m/s | 30 cm |
Test tasks.
1 (A) A material point is:
1) a body of negligible mass;
2) the body is very small;
3) a point showing the position of the body in space;
4) a body whose dimensions can be neglected in the conditions of this problem.
2(A) What is the change in position of one body relative to another called:
1) trajectory;
2) moving;
4) mechanical movement.
3(A) What is the displacement of a point moving in a circle of radius R when it rotates 180º?
1) R/2 2) R 3) 2R 4) R
4(A) The line that a body describes when moving in space is called:
1) trajectory;
2) moving;
4) mechanical movement.
5(A)
The figure shows a graph of the movement of a body from point A to point B and back. Point A is located at point x 0 = 30 m, and point B is at point x = 5 m. What is the minimum speed of the bus along the entire route there and back?
1) 32 m/s 2) 23 m/s 3) 20 m/s 4) 21.5 m/s
10(A) A body that begins to move uniformly accelerated from a state of rest with an acceleration of 2 m/s 2, then in the third second it will cover the distance
1) 7 m 2) 5 m 3) 3 m 4) 2 m
11(A) The coordinates of bodies A and B moving along the same straight line change over time, as shown in the graph. What is the speed of body A relative to body B?
1) 40 m/s x, m
12(A) The escalator staircase rises up with a speed u, at what speed relative to the walls should a person go down it in order to rest relative to the people standing on the stairs going down?
1) u 2) 2u 3) 3u 4) 4u
13(A) At a speed of 12 m/s, the braking time of a truck is 4 s. If, when braking, the acceleration of the car is constant and does not depend on the initial speed, then when braking, the car will reduce its speed from 18 m/s to 15 m/s, having passed
1) 12.3 m 3) 28.4 m
2) 16.5 m 4) 33.4 m
14(A) A truck and a motorcyclist are traveling along a 5 km long ring road in one direction at speeds u 1, respectively. = 40 km/h u 2 = 100 km/h. If at the initial moment of time they were in the same place, then the motorcyclist will catch up with the car, passing
1) 3.3 km 3) 8.3 km
2) 6.2 km 4) 12.5 km
15(A) A body was thrown from the Earth's surface at an angle α to the horizon with an initial speed u 0 = 10 m/s, if the body’s flight range is L = 10 m, then the angle α is equal to
1) 15º 2) 22.5º 3) 30º 4) 45º
16(A) A boy threw a ball horizontally from a window located at a height of 20 m. The ball fell at a distance of 8 m from the wall of the house. At what initial speed was the ball thrown?
1) 0.4 m/s 2) 2.5 m/s 3) 3 m/s 4) 4 m/s
17(B) A material point moves at a constant speed along a circle of radius R. How will the physical quantities listed in the first column change if the speed of the point increases?
Physical quantities. Their change.
A) Angular velocity 1) will increase
B) Centripetal 2) will decrease
acceleration 3) will not change
B) Circulation period
circumferentially
| A | B | IN |
18(B)
Using the graph of the body's speed as a function of time, determine the distance traveled in 5 s.
υ, m/s
19(B) Centripetal acceleration of a material point moving in a circle when the linear speed increases by 2 times and angular velocity 2 times with a constant radius increased by .... once.
20(C) An inclined plane intersects a horizontal plane along straight line AB.
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Problem 1. Two small steel balls are thrown simultaneously from the same point from the surface of the earth with initial velocities u01 = 5 m/s and v02 = 8 m/s, directed at angles ", = 80° and a2 = 20° to the horizon, respectively . What is the distance between the balls after time / = -^s after the throw? The trajectories of the balls lie in the same vertical plane. Neglect air resistance. Solution. The balls move in the Earth's gravitational field with constant acceleration g (we neglect air resistance). Let us choose a coordinate system as shown in Fig. 20, we place the starting point at the throwing point. For radius vectors, balls. Let's choose a coordinate system. The required distance. Projection of acceleration The required distance / is equal to the modulus of the difference between the radius vectors of the balls at the moment of time / = - s. Since the balls were thrown from the same point, then /*0| = r02, therefore: / = . (The remaining terms were destroyed when subtracting the radius-vectopes.) In turn, according to the cosine theorem (see Fig. 20): Substituting into this equality numeric values of the quantities included in it, we obtain \v0l -v02\ = 7 m/s. Then the required distance between the balls at the moment of time * Problem 2. Two bodies are thrown vertically upward from the surface of the earth from one point following each other with a time interval r, with the same initial velocities v0. Neglecting air resistance, determine how long after they “meet”? Please comment on the solution for Solution. Let's direct the Oy axis vertically upward, placing the origin of reference at the throwing point. We will count down time starting from the moment the first body is thrown. Initial conditions of motion of bodies: O "o = = 0, vy0l = v0; 2) t0 = r, y02 = O, vy02 = v0. Projections of acceleration of bodies in the absence of air resistance are equal: avl = ay2 = -g. Equations of motion of bodies in projections onto the Oy axis taking into account initial conditions have the form: (Note that y2 = O at 0 For clarity, let us depict the graphs of these functions in one drawing (Fig. 21). From the drawing it is clear that the “meeting” will occur at some point in time at point A, where the graphs yx(t Thus, ^^ the “meeting” condition: y, (O = Vr (A) "that is, = v0 ft -r) 2 "2 Solving this equation for /v, we find: tx = - + -. Let's analyze by - g 2 obtained expression for It is known (see Example 7) that the flight time of a body thrown vertically is equal to 2v0/g. Therefore, if v0 2v0/g. This means that the first body will fall to the ground first, and only then the second will be thrown up. In other words, the bodies will “meet” at the throwing point. Problem 3. A boy, being on a flat mountain slope with an angle of inclination (p- 30°), throws a stone towards the rise of the mountain, giving it an initial speed v0 directed at an angle /? = 60° to the horizon. At what distance from the boy will the stone fall ? Neglect air resistance. Solution. Let us choose a reference system as shown in Fig. 22, placing the origin O at the throwing point. In this reference system starting speed The stone makes an angle a = ft-(p = 30°) with the Ox axis. Initial conditions: Fig. 22 Projections of acceleration of the stone in the absence of air resistance are equal (see Fig. 22): ax = gx = -gsin#?, ау =gy = -g Here we took into account that the angle between the vector g and the perpendicular to the surface of the mountain equal to angle slope of the mountain (р- 30° (why?), in addition, according to the conditions of the problem (р = а. Let us write the equations of system (14) taking into account the initial conditions: t2 Г x(t) = (y0cos«)/-(gsin^ >)-, y(t) = (v0sina)t-(gcosp)-. We find the flight time r of the stone from the last equation, knowing that We choose a coordinate system. The required distance. Acceleration projection Namely r = -=-. (Value We discarded g = 0, since it is not related to the problem. Substituting the found value of g into the equation for g(/), we determine the required distance (in other words, flight range): 3 g Problem 4 The massive platform moves at a constant speed K0 along the horizontal floor. A ball is hit from the rear edge of the platform. The modulus of the initial velocity of the ball relative to the platform is equal to y\ u = 2VQ9, and the vector u makes an angle a = 60° with the horizon (Fig. 23). To what maximum height above the floor will the ball rise? At what distance from the edge of the platform will the ball be at the moment _ j. w_ ,0 of landing. Neglect the height of the platform and air resistance. All velocities lie in the same vertical plane. (FZFTSH at MIPT, 2009.) Solution. To describe the movement of the ball and the platform, we introduce a reference system associated with the floor. Let's direct the Ox axis horizontally in the direction of impact, and the Oy axis vertically upward (Fig. 23). The ball moves with constant acceleration a, with ax = 0, aY = -g, where g is the magnitude of the acceleration of free fall. The projections of the initial speed v0 of the ball on the Ox and Oy axes are equal: v0,x = V0, + = -K + 2F0 cos 60° = -V0 + V0 = 0, % = K, - + =10 + sin 60° = >/ 3F0. If the ball's horizontal velocity is zero, it means that it only moves vertically and will fall at the point of impact. We will find the maximum lifting height (ynvix) and the flight time of the ball from the laws of kinematics of uniformly accelerated motion: a/ Select a coordinate system. The required distance. Projection of acceleration Zt Considering that at y = y^ the projection of vertical velocity becomes zero vY = 0, and at the moment of landing of the ball t = Gflight its coordinate along the Oy axis becomes zero y = 0, we have: ZU-t = 1 flight 2 g 2 g - S During the flight of the ball, the platform will shift by a flight distance 8 U sh which is the desired distance between the ball and the platform at the moment the ball lands. Test questions 1. In Fig. Figure 24 shows the trajectory of the body. His starting position is designated by point A, the final point - by point C. What are the projections of the displacement of the body on the Ox and Oy axes, the module of displacement and the path traveled by the body? 2. The body moves uniformly and rectilinearly on xOy plane. Its coordinates change depending on time in accordance with the equations: (values are measured in SI). Write down the equation y = y(x) for the trajectory of the body. What are they equal to? initial coordinates body and its coordinates 2 s after the start of movement? 3. Rod AB, oriented along the Ox axis, moves at a constant speed v = 0.1 m/s in the positive direction of the axis. The front end of the rod is point A, the rear end is point B. What is the length of the rod if at time tA = 1 °C after the start of movement the coordinate of point A is equal to x, = 3m, and at time tB- 30s the coordinate of point B is *L =4.5m? (MIET, 2006) 4. When two bodies move, how is their relative speed determined? 5. A bus and a motorcycle are located at a distance of L = 20 km from each other. If they move in the same direction with certain speeds r\ and v2, respectively, then the motorcycle will catch up with the bus in time / = 1 hour. What is the speed of the motorcycle relative to the bus? 6. What is called average ground speed bodies? 7. The first hour of the journey the train traveled at a speed of 50 km/h, the next 2 hours it traveled at a speed of 80 km/h. Find the average speed of the train during these 3 hours. Select correct option answer and justify your choice: 1) 60 km/h; 2) 65 km/h; 3) 70 km/h; 4) 72 km/h; 5) 75 km/h. (RGTU named after K. E. Tsiolkovsky (MATI), 2006) 8. One-fifth of the way the car was traveling at a speed r\ = 40 km/h, and the rest of the way at a speed v2 = 60 km/h. Find the average speed of the car along the entire route. (MEPhI, 2006) 9. The material point begins to move along the Ox axis according to the law *(/) = 5 + 4/-2r(m). At what distance from the origin will the speed of the point be zero? (MSTU named after N. E. Bauman, 2006) 10. The skater, having accelerated to a speed v0 = 5 m/s, began to slide straight and equally slow. After time t = 20 s, the speed module of the skater became equal to v = 3 m/s. What is the speed skater's acceleration? Problems 1. A pedestrian ran for a third of the entire journey at a speed v( =9 km/h, a third of the entire time walked at a speed v2 =4 km/h, and the rest of the time walked at a speed equal to the average speed along the entire path. Find this speed. (ZFTSH at MIPT, 2001) 2. A body, moving uniformly accelerated and rectilinearly from a state of rest, covered a distance S in time r. What speed did the body have at the moment when it passed the distance S/n, where n is some positive number? (MEPhI, 2006) 3. A body falls without an initial speed and reaches the surface of the earth after 4 s. From what height did the body fall? Neglect air resistance. Choose the correct answer and justify your choice: 1) 20m; 2) 40 m; 3) 80m; 4) 120m; 5) 160 m. (RGTU named after K. E. Tsiolkovsky (MATI), 2006) 4. A stone thrown vertically upward from the surface of the earth fell to the ground after T = 2s. Determine the distance 5 traveled by the stone in time r = 1.5 s after being thrown. Neglect air resistance. Acceleration of free fall is taken equal to g = 10 m/s2. (MIET, 2006) Let's choose a coordinate system. The required distance. Projection of acceleration 5. From one point at a height h from the surface of the earth are thrown with at the same speeds stone A vertically upward and stone B vertically downward. It is known that stone A reached the top point of its trajectory at the same time that stone B fell to the ground. Which maximum height(counting from the surface of the earth) reached stone A? Ignore air resistance. (MIPT, 1997) 6. A stone is thrown horizontally from a mountain slope forming an angle a = 45° with the horizon (Fig. 25). What is the initial speed v0 of the stone if it fell onto a slope at a distance / = 50 m from the point of throwing? Neglect air resistance. 7. A body is thrown horizontally. 3 s after the throw, the angle between the direction of full speed and the direction of full acceleration became equal to 60°. Determine the total speed of the body at this moment in time. Neglect air resistance. (RSU of Oil and Gas named after I.M. Gubkin, 2006) Instruction. By full speed and full acceleration we simply mean the speed and acceleration of a body. 8. The shell exploded into several fragments that flew in all directions at the same speeds. The fragment, flying vertically down, reached the ground in time. The fragment, flying vertically upward, fell to the ground after time t2. How long did it take for the fragments that flew horizontally to fall? Ignore air resistance. (MIPT, 1997) 9. A stone thrown at an angle to the horizon reached greatest height 5 m. Find full time flight of stone. Neglect air resistance. (RSU of Oil and Gas named after I.M. Gubkin, 2006) 10. A stone thrown from the surface of the earth at an angle a = 30° to the horizon twice reached the same height h after time = 3s and = 5s after the start of movement. Find the initial speed of the stone v0. Acceleration of free fall is taken equal to g = 10 m/s2. Neglect air resistance. (Institute of Cryptography, Communications and Informatics of the Academy of the Federal Security Service of the Russian Federation, 2006) 11. At what speed v0 should a projectile fly out of a cannon at the moment of launch of the rocket in order to shoot it down? The rocket launches vertically with constant acceleration i = 4 m/s2. The distance from the gun to the rocket launch site (they are on the same horizontal level) is equal to / = 9 km. The cannon fires at an angle « = 45° to the horizontal. Neglect air resistance.