Motion of a body with constant acceleration. Acceleration

§ 12th. Movement with constant acceleration

At uniformly accelerated motion The following equations are valid, which we present without derivation:

As you understand, vector formula on the left and two scalar formulas on the right are equal. From an algebraic point of view, scalar formulas mean that with uniformly accelerated motion, the displacement projections depend on time according to a quadratic law. Compare this with the nature of instantaneous velocity projections (see § 12-h).

Knowing that s x = x – x o And s y = y – y o(see § 12th), of the two scalar formulas from the upper right column we get equations for coordinates:

Since the acceleration during uniformly accelerated motion of a body is constant, then coordinate axes can always be positioned so that the acceleration vector is directed parallel to one axis, for example the Y axis. Consequently, the equation of motion along the X axis will be noticeably simplified:

x = x o + υ ox t + (0) And y = y o + υ oy t + ½ a y t²

Please note that the left equation coincides with the equation of uniform rectilinear motion (see § 12-g). It means that uniformly accelerated motion can “add up” from uniform motion along one axis and uniformly accelerated motion along the other. This is confirmed by the experience with the core on a yacht (see § 12-b).

Task. Stretching out her arms, the girl tossed the ball. He rose 80 cm and soon fell at the girl’s feet, flying 180 cm. At what speed was the ball thrown and what speed did the ball have when it hit the ground?

Let's square both sides of the equation to project the instantaneous velocity onto the Y axis: υ y = υ oy + a y t(see § 12). We get the equality:

υ y ² = ( υ oy + a y t )² = υ oy ² + 2 υ oy a y t + a y ² t²

Let's take the factor out of brackets 2 a y only for the two right-hand terms:

υ y ² = υ oy ² + 2 a y ( υ oy t + ½ a y t² )

Note that in brackets we get the formula for calculating the displacement projection: s y = υ oy t + ½ a y t². Replacing it with s y, we get:

Solution. Let's make a drawing: direct the Y axis upward, and place the origin of coordinates on the ground at the girl's feet. Let us apply the formula we derived for the square of the velocity projection, first at the top point of the ball’s rise:

0 = υ oy ² + 2·(–g)·(+h) ⇒ υ oy = ±√¯2gh = +4 m/s

Then, when starting to move from the top point down:

υ y² = 0 + 2·(–g)·(–H) ⇒ υ y = ±√¯2gh = –6 m/s

Answer: the ball was thrown upward with a speed of 4 m/s, and at the moment of landing it had a speed of 6 m/s, directed against the Y axis.

Note. We hope you understand that the formula for the square of the projection of instantaneous velocity will be correct by analogy for the X axis.

Lesson objectives:

Educational:

Vos nutritious

Lesson type : Combined lesson.

View document contents
“Lesson topic: “Acceleration. Rectilinear motion with constant acceleration."

Prepared by Marina Nikolaevna Pogrebnyak, physics teacher at MBOU “Secondary School No. 4”

Class -11

Lesson 5/4 Lesson topic: “Acceleration. Straight-line movement with constant acceleration».

Lesson objectives:

Educational: Introduce students to characteristic features rectilinear uniformly accelerated motion. Give the concept of acceleration as the main physical quantity characterizing uneven movement. Enter a formula to determine the instantaneous speed of a body at any time, calculate the instantaneous speed of a body at any time,

improve students' ability to solve problems analytically and graphically.

Educational: development of schoolchildren's theoretical, creative thinking, formation of operational thinking aimed at choosing optimal solutions

Vosnutritious : bring up conscious attitude to study and interest in studying physics.

Lesson type : Combined lesson.

Demos:

1. Uniformly accelerated motion of the ball along inclined plane.

2. Multimedia application “Fundamentals of Kinematics”: fragment “Uniformly accelerated motion”.

Progress.

1.Organizational moment.

2. Test of knowledge: Independent work(“Movement.” “Graphs of rectilinear uniform motion”) - 12 min.

3. Studying new material.

Plan for presenting new material:

1. Instantaneous speed.

2. Acceleration.

3. Speed during rectilinear uniformly accelerated motion.

1. Instantaneous speed. If the speed of a body changes with time, to describe the movement you need to know what the speed of the body is at this moment time (or at a given point in the trajectory). This speed is called instantaneous speed.

It can also be said that instantaneous speed is the average speed over a very short time interval. When driving at a variable speed, the average speed measured over different time intervals will be different.

However, if when measuring average speed take smaller and smaller time intervals, the value of the average speed will tend to a certain a certain value. This is the instantaneous speed at a given moment in time. In the future, when speaking about the speed of a body, we will mean its instantaneous speed.

2. Acceleration. With uneven movement, the instantaneous speed of a body is a variable quantity; it is different in modulus and (or) in direction different moments time and in different points trajectories. All speedometers of cars and motorcycles show us only the instantaneous speed module.

If the instantaneous speed of uneven motion changes unequally over equal periods of time, then it is very difficult to calculate it.

Such complex uneven movements are not studied at school. Therefore, we will consider only the simplest non-uniform motion - uniformly accelerated rectilinear motion.

Rectilinear motion, in which the instantaneous speed changes equally over any equal time intervals, is called uniformly accelerated rectilinear motion.

If the speed of a body changes during movement, the question arises: what is the “rate of change of speed”? This quantity, called acceleration, plays vital role in all mechanics: we will soon see that the acceleration of a body is determined by the forces acting on this body.

Acceleration is the ratio of the change in the speed of a body to the time interval during which this change occurred.

The SI unit of acceleration is m/s2.

If a body moves in one direction with an acceleration of 1 m/s 2 , its speed changes by 1 m/s every second.

The term "acceleration" is used in physics when talking about any change in speed, including when the velocity modulus decreases or when the velocity modulus remains unchanged and the speed changes only in direction.

3. Speed during rectilinear uniformly accelerated motion.

From the definition of acceleration it follows that v = v 0 + at.

If we direct the x axis along the straight line along which the body moves, then in projections onto the x axis we obtain v x = v 0 x + a x t.

Thus, with rectilinear uniformly accelerated motion, the projection of velocity depends linearly on time. This means that the graph of v x (t) is a straight line segment.

Movement formula:

Speed graph of an accelerating car:

Speed graph of a braking car

4. Consolidation of new material.

What is the instantaneous speed of a stone thrown vertically upward at the top point of its trajectory?

About what speed - average or instantaneous - we're talking about in the following cases:

a) the train traveled between stations at a speed of 70 km/h;

b) the speed of movement of the hammer upon impact is 5 m/s;

c) the speedometer on the electric locomotive shows 60 km/h;

d) a bullet leaves a rifle at a speed of 600 m/s.

TASKS SOLVED IN THE LESSON

The OX axis is directed along the trajectory of the rectilinear motion of the body. What can you say about the movement in which: a) v x 0, and x 0; b) v x 0, a x v x x 0;

d) v x x v x x = 0?

1. A hockey player lightly hit the puck with his stick, giving it a speed of 2 m/s. What will be the speed of the puck 4 s after impact if, as a result of friction with ice, it moves with an acceleration of 0.25 m/s 2?

2. The train, 10 s after the start of movement, acquires a speed of 0.6 m/s. How long after the start of movement will the speed of the train become 3 m/s?

5. HOMEWORK: §5,6, ex. 5 No. 2, ex. 6 No. 2.

Among the various movements with constant acceleration, the simplest is rectilinear movement. If at the same time the velocity module increases, then the movement is sometimes called uniformly accelerated, and when the velocity module decreases, it is called uniformly decelerated. This kind of movement is made by a train departing from or approaching a station. A stone thrown vertically downwards moves equally accelerated, and a stone thrown vertically upwards moves equally slowly.
To describe rectilinear motion with constant acceleration, you can use one coordinate axis (for example, the X axis), which is expediently directed along the motion trajectory. In this case, any problem is solved using two equations:
(1.20.1)

And
2? Projection of displacement and path during rectilinear motion with constant acceleration We find the projection on the X-axis of displacement, equal to Ax = x - x0, from equation (1.20.2):
M2
Ax = v0xt +(1.20.3)
If the speed of the body (point) does not change its direction, then the path equal to modulus displacement projections
.2
s = |Ax| =
(1.20.4)
axt
VoJ + -o
If the speed changes its direction, then the path is more difficult to calculate. In this case, it consists of the displacement module up to the moment of changing the direction of speed and the displacement module after this moment.
Average speed during straight-line motion with constant acceleration
From formula (1.19.1) it follows that
+ ^ = Ax 2 t "
Oh
But - is the projection of the average speed onto the X axis (see § 1.12),
i.e. ^ = v. Consequently, with rectilinear motion from t
With constant acceleration, the projection of the average speed onto the X axis is equal to:
!)ag + Vr
vx= 0x2 . (1.20.5)
It can be proven that if some other physical quantity is in linear dependence from time, then the time average value of this quantity is equal to half the sum of its smallest and highest values during a given period of time.
If during rectilinear motion with constant acceleration the direction of velocity does not change, then the average velocity module is equal to half the sum of the initial and final speed, i.e.
K* + vx\ v0 + v
Relationship between projections of initial and final velocities, acceleration and displacement
According to formula (1.19.1)
Lx = °*2 xt. (1.20.7)
Time t can be expressed from formula (1.20.1)
Vx~V0x ah
and substitute into (1.20.7). We get:
Vx + V0x Vx - v0x V2X - i>jj
= 2 ST" --257-
From here
v2x = v Іх+2а3Лх. (1.20.8)
It is useful to remember formula (1.20.8) and expression (1.20.6) for average speed. These formulas may be needed to solve many problems.
? 1. What is the direction of acceleration when the train departs from the station (acceleration)? When approaching a station (braking)?
Draw a graph of the path during acceleration and during braking.
Prove yourself that in uniformly accelerated rectilinear motion without initial speed ways, traversable by the body for equal successive intervals of time, proportional to successive odd numbers:
Sj: S2* Sg ... = 1: 3: 5: ... . This was first proven by Galileo.

Motion of a body with constant acceleration. Acceleration

View document contents “Lesson topic: “Acceleration. Rectilinear motion with constant acceleration."

More on the topic §1.20. STRAIGHT LINEAR MOTION WITH CONSTANT ACCELERATION:

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“Lesson topic: “Acceleration. Rectilinear motion with constant acceleration."