Acceleration- a physical vector quantity that characterizes how quickly a body (material point) changes the speed of its movement. Acceleration is an important kinematic characteristic of a material point.
The simplest type of motion is uniform motion in a straight line, when the speed of the body is constant and the body covers the same path in any equal intervals of time.
But most movements are uneven. In some areas the body speed is greater, in others less. As the car begins to move, it moves faster and faster. and when stopping it slows down.
Acceleration characterizes the rate of change in speed. If, for example, the acceleration of a body is 5 m/s 2, then this means that for every second the speed of the body changes by 5 m/s, i.e. 5 times faster than with an acceleration of 1 m/s 2.
If the speed of a body during uneven motion changes equally over any equal periods of time, then the motion is called uniformly accelerated.
The SI unit of acceleration is the acceleration at which for every second the speed of the body changes by 1 m/s, i.e. meter per second per second. This unit is designated 1 m/s2 and is called “meter per second squared”.
Like speed, the acceleration of a body is characterized not only by its numerical value, but also by its direction. This means that acceleration is also a vector quantity. Therefore, in the pictures it is depicted as an arrow.
If the speed of a body during uniformly accelerated linear motion increases, then the acceleration is directed in the same direction as the speed (Fig. a); if the speed of the body decreases during a given movement, then the acceleration is directed in the opposite direction (Fig. b).
Average and instantaneous acceleration
The average acceleration of a material point over a certain period of time is the ratio of the change in its speed that occurred during this time to the duration of this interval:
\(\lt\vec a\gt = \dfrac (\Delta \vec v) (\Delta t) \)
The instantaneous acceleration of a material point at some point in time is the limit of its average acceleration at \(\Delta t \to 0\) . Keeping in mind the definition of the derivative of a function, instantaneous acceleration can be defined as the derivative of speed with respect to time:
\(\vec a = \dfrac (d\vec v) (dt) \)
Tangential and normal acceleration
If we write the speed as \(\vec v = v\hat \tau \) , where \(\hat \tau \) is the unit unit of the tangent to the trajectory of motion, then (in a two-dimensional coordinate system):
\(\vec a = \dfrac (d(v\hat \tau)) (dt) = \)
\(= \dfrac (dv) (dt) \hat \tau + \dfrac (d\hat \tau) (dt) v =\)
\(= \dfrac (dv) (dt) \hat \tau + \dfrac (d(\cos\theta\vec i + sin\theta \vec j)) (dt) v =\)
\(= \dfrac (dv) (dt) \hat \tau + (-sin\theta \dfrac (d\theta) (dt) \vec i + cos\theta \dfrac (d\theta) (dt) \vec j))v\)
\(= \dfrac (dv) (dt) \hat \tau + \dfrac (d\theta) (dt) v \hat n \),
where \(\theta \) is the angle between the velocity vector and the x-axis; \(\hat n \) - unit unit perpendicular to the speed.
Thus,
\(\vec a = \vec a_(\tau) + \vec a_n \),
Where \(\vec a_(\tau) = \dfrac (dv) (dt) \hat \tau \)- tangential acceleration, \(\vec a_n = \dfrac (d\theta) (dt) v \hat n \)- normal acceleration.
Considering that the velocity vector is directed tangent to the trajectory of motion, then \(\hat n \) is the unit unit of the normal to the trajectory of motion, which is directed to the center of curvature of the trajectory. Thus, normal acceleration is directed towards the center of curvature of the trajectory, while tangential acceleration is tangential to it. Tangential acceleration characterizes the rate of change in the magnitude of velocity, while normal acceleration characterizes the rate of change in its direction.
Movement along a curved trajectory at each moment of time can be represented as rotation around the center of curvature of the trajectory with angular velocity \(\omega = \dfrac v r\) , where r is the radius of curvature of the trajectory. In this case
\(a_(n) = \omega v = (\omega)^2 r = \dfrac (v^2) r \)
Acceleration measurement
Acceleration is measured in meters (divided) per second to the second power (m/s2). The magnitude of the acceleration determines how much the speed of a body will change per unit time if it constantly moves with such acceleration. For example, a body moving with an acceleration of 1 m/s 2 changes its speed by 1 m/s every second.
Acceleration units
- meter per second squared, m/s², SI derived unit
- centimeter per second squared, cm/s², derived unit of the GHS system
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If the instantaneous speed of a moving body increases, then the movement is called accelerated; if the instantaneous speed decreases, then the movement is called slow.
Speed changes differently in different uneven movements. For example, a freight train, leaving the station, moves at an accelerated pace; on the stretch - sometimes accelerated, sometimes evenly, sometimes slowly; approaching the station, he moves slowly. A passenger train also moves unevenly, but its speed changes faster than that of a freight train. The speed of a bullet in the bore of a rifle increases from zero to hundreds of meters per second in a few thousandths of a second; when hitting an obstacle, the bullet's speed decreases to zero very quickly. When a rocket takes off, its speed increases slowly at first, and then more and more quickly.
Among the various accelerated movements, there are movements in which the instantaneous speed for any equal periods of time increases by the same amount. Such movements are called uniformly accelerated. A ball that begins to roll down an inclined plane or begins to freely fall to the Earth moves with uniform acceleration. Note that the uniformly accelerated nature of this movement is disrupted by friction and air resistance, which we will not take into account for now.
The greater the angle of inclination of the plane, the faster the speed of the ball rolling along it increases. The speed of a freely falling ball increases even faster (by about 10 m/s for every second). For uniformly accelerated motion, it is possible to quantitatively characterize the change in speed over time by introducing a new physical quantity - acceleration.
In the case of uniformly accelerated motion, acceleration is the ratio of the increment in speed to the period of time during which this increment occurred:
We will denote acceleration by the letter . Comparing with the corresponding expression from § 9, we can say that acceleration is the rate of change of velocity.
Suppose that at the moment of time the speed was , and at the moment it became equal, so that over time the increment in speed is . This means acceleration
(16.1)
From the definition of uniformly accelerated motion it follows that this formula will give the same acceleration, no matter what time period you choose. From here it is also clear that with uniformly accelerated motion, acceleration is numerically equal to the increment in speed per unit time. The SI unit of acceleration is meter per second squared (m/s2), i.e. meter per second per second.
If the path and time are measured in other units, then for acceleration it is necessary to take the corresponding units of measurement. No matter in what units the path and time are expressed, in the designation of the unit of acceleration the unit of length is in the numerator, and the square of the unit of time is in the denominator. The rule for moving to other units of length and time for acceleration is similar to the rule for speeds (§11). For example,
1 cm/s^2=36 m/min^2.
If the motion is not uniformly accelerated, then we can introduce, using the same formula (16.1), the concept of average acceleration. It characterizes the change in speed over a certain period of time along the section of the route covered during this period of time. On individual segments of this section, the average acceleration can have different values (cf. what was said in § 14).
If we choose such small time intervals that within each of them the average acceleration remains practically unchanged, then it will characterize the change in speed over any part of this interval. The acceleration found in this way is called instantaneous acceleration (usually the word “instantaneous” is omitted, cf. § 15). In uniformly accelerated motion, the instantaneous acceleration is constant and equal to the average acceleration over any period of time.
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In this lesson, we will look at an important characteristic of uneven motion - acceleration. In addition, we will consider uneven motion with constant acceleration. Such movement is also called uniformly accelerated or uniformly decelerated. Finally, we will talk about how to graphically depict the dependence of the speed of a body on time during uniformly accelerated motion.
Homework
Having solved the problems for this lesson, you will be able to prepare for questions 1 of the State Examination and questions A1, A2 of the Unified State Exam.
1. Problems 48, 50, 52, 54 sb. problems A.P. Rymkevich, ed. 10.
2. Write down the dependence of speed on time and draw graphs of the dependence of the speed of the body on time for the cases shown in Fig. 1, cases b) and d). Mark turning points on the graphs, if any.
3. Consider the following questions and their answers:
Question. Is the acceleration due to gravity an acceleration as defined above?
Answer. Of course it is. The acceleration of gravity is the acceleration of a body that is freely falling from a certain height (air resistance must be neglected).
Question. What will happen if the acceleration of the body is directed perpendicular to the speed of the body?
Answer. The body will move uniformly around the circle.
Question. Is it possible to calculate the tangent of an angle using a protractor and a calculator?
Answer. No! Because the acceleration obtained in this way will be dimensionless, and the dimension of acceleration, as we showed earlier, should have the dimension m/s 2.
Question. What can be said about motion if the graph of speed versus time is not straight?
Answer. We can say that the acceleration of this body changes with time. Such a movement will not be uniformly accelerated.
Acceleration is a quantity that characterizes the rate of change in speed.
For example, when a car starts moving, it increases its speed, that is, it moves faster. At first its speed is zero. Once moving, the car gradually accelerates to a certain speed. If a red traffic light comes on on its way, the car will stop. But it will not stop immediately, but over time. That is, its speed will decrease down to zero - the car will move slowly until it stops completely. However, in physics there is no term “slowdown”. If a body moves, slowing down its speed, then this will also be an acceleration of the body, only with a minus sign (as you remember, speed is a vector quantity).
> is the ratio of the change in speed to the period of time during which this change occurred. The average acceleration can be determined by the formula:
Rice. 1.8. Average acceleration. In SI acceleration unit– is 1 meter per second per second (or meter per second squared), that is
A meter per second squared is equal to the acceleration of a point moving in a straight line, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.
Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity equal to the limit to which the average acceleration tends as the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:
With accelerated linear motion, the speed of the body increases in absolute value, that is
V 2 > v 1
and the direction of the acceleration vector coincides with the velocity vector
If the speed of a body decreases in absolute value, that is
V 2< v 1
then the direction of the acceleration vector is opposite to the direction of the velocity vector. In other words, in this case what happens is slowing down, in this case the acceleration will be negative (and< 0). На рис. 1.9 показано направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.
Rice. 1.9. Instant acceleration.
When moving along a curved path, not only the speed module changes, but also its direction. In this case, the acceleration vector is represented as two components (see next section).
Tangential (tangential) acceleration– this is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the movement trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.
Rice. 1.10. Tangential acceleration.
The direction of the tangential acceleration vector (see Fig. 1.10) coincides with the direction of linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis with the tangent circle, which is the trajectory of the body.
Normal acceleration
Normal acceleration is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in direction and is denoted by the letter. The normal acceleration vector is directed along the radius of curvature of the trajectory.
Full acceleration
Full acceleration during curvilinear motion, it consists of tangential and normal accelerations along and is determined by the formula:
(according to the Pythagorean theorem for a rectangular rectangle).