In this topic we will look at a very special type of irregular motion. Based on the opposition to uniform motion, uneven motion is movement at unequal speed along any trajectory. What is the peculiarity of uniformly accelerated motion? This is an uneven movement, but which "equally accelerated". We associate acceleration with increasing speed. Let's remember the word "equal", we get an equal increase in speed. How do we understand “equal increase in speed”, how can we evaluate whether the speed is increasing equally or not? To do this, we need to record time and estimate the speed over the same time interval. For example, a car starts to move, in the first two seconds it develops a speed of up to 10 m/s, in the next two seconds it reaches 20 m/s, and after another two seconds it already moves at a speed of 30 m/s. Every two seconds the speed increases and each time by 10 m/s. This is uniformly accelerated motion.

The physical quantity that characterizes how much the speed increases each time is called acceleration.

Can the movement of a cyclist be considered uniformly accelerated if, after stopping, in the first minute his speed is 7 km/h, in the second - 9 km/h, in the third - 12 km/h? It is forbidden! The cyclist accelerates, but not equally, first he accelerated by 7 km/h (7-0), then by 2 km/h (9-7), then by 3 km/h (12-9).

Typically, movement with increasing speed is called accelerated movement. Movement with decreasing speed is slow motion. But physicists call any movement with changing speed accelerated movement. Whether the car starts moving (the speed increases!) or brakes (the speed decreases!), in any case it moves with acceleration.

Uniformly accelerated motion- this is the movement of a body in which its speed for any equal intervals of time changes(can increase or decrease) the same

Body acceleration

Acceleration characterizes the rate of change in speed. This is the number by which the speed changes every second. If the acceleration of a body is large in magnitude, this means that the body quickly gains speed (when it accelerates) or quickly loses it (when braking). Acceleration is a physical vector quantity, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.

Let's determine the acceleration in the next problem. At the initial moment of time, the speed of the ship was 3 m/s, at the end of the first second the speed of the ship became 5 m/s, at the end of the second - 7 m/s, at the end of the third 9 m/s, etc. Obviously, . But how did we determine? We are looking at the speed difference over one second. In the first second 5-3=2, in the second second 7-5=2, in the third 9-7=2. But what if the speeds are not given for every second? Such a problem: the initial speed of the ship is 3 m/s, at the end of the second second - 7 m/s, at the end of the fourth 11 m/s. In this case, you need 11-7 = 4, then 4/2 = 2. We divide the speed difference by the time period.

This formula is most often used in a modified form when solving problems:

The formula is not written in vector form, so we write the “+” sign when the body is accelerating, the “-” sign when it is slowing down.

Acceleration vector direction

The direction of the acceleration vector is shown in the figures

In this figure, the car moves in a positive direction along the Ox axis, the velocity vector always coincides with the direction of movement (directed to the right). When the acceleration vector coincides with the direction of the speed, this means that the car is accelerating. Acceleration is positive.

During acceleration, the direction of acceleration coincides with the direction of speed. Acceleration is positive.

In this picture, the car is moving in the positive direction along the Ox axis, the velocity vector coincides with the direction of movement (directed to the right), the acceleration does NOT coincide with the direction of the speed, this means that the car is braking. Acceleration is negative.

When braking, the direction of acceleration is opposite to the direction of speed. Acceleration is negative.

Let's figure out why the acceleration is negative when braking. For example, in the first second the motor ship dropped its speed from 9m/s to 7m/s, in the second second to 5m/s, in the third to 3m/s. The speed changes to "-2m/s". 3-5=-2; 5-7=-2; 7-9=-2m/s. This is where it comes from negative meaning acceleration.

When solving problems, if the body slows down, acceleration is substituted into the formulas with a minus sign!!!

Moving during uniformly accelerated motion

An additional formula called timeless

Formula in coordinates

Medium speed communication

With uniformly accelerated motion, the average speed can be calculated as the arithmetic mean of the initial and final speeds

From this rule follows a formula that is very convenient to use when solving many problems

Path ratio

If a body moves uniformly accelerated, the initial speed is zero, then the paths traversed in successive equal intervals of time are related as a successive series of odd numbers.

The main thing to remember

1) What is uniformly accelerated motion;
2) What characterizes acceleration;
3) Acceleration is a vector. If a body accelerates, the acceleration is positive, if it slows down, the acceleration is negative;
3) Direction of the acceleration vector;
4) Formulas, units of measurement in SI

Exercises

Two trains are moving towards each other: one is heading north at an accelerated rate, the other is moving slowly to the south. How are train accelerations directed?

Equally to the north. Because the first train's acceleration coincides in direction with the movement, and the second train's acceleration is opposite to the movement (it slows down).

Acceleration in kinematics formula. Acceleration in kinematics definition.

What is acceleration?

Speed ​​may change while driving.

Velocity is a vector quantity.

The velocity vector can change in direction and magnitude, i.e. in size. To account for such changes in speed, acceleration is used.

Acceleration definition

Definition of acceleration

Acceleration is a measure of any change in speed.

Acceleration, also called total acceleration, is a vector.

Acceleration vector

The acceleration vector is the sum of two other vectors. One of these other vectors is called tangential acceleration, and the other is called normal acceleration.

Describes the change in the magnitude of the velocity vector.

Describes the change in direction of the velocity vector.

At straight motion the direction of speed does not change. In this case, the normal acceleration is zero, and the total and tangential accelerations coincide.

With uniform motion, the velocity module does not change. In this case, the tangential acceleration is zero, and the total and normal accelerations are the same.

If a body performs rectilinear uniform motion, then its acceleration is zero. And this means that the components of total acceleration, i.e. normal acceleration and tangential acceleration are also zero.

Full acceleration vector

The total acceleration vector is equal to the geometric sum of the normal and tangential accelerations, as shown in the figure:

Acceleration formula:

a = a n + a t

Full acceleration module

Full acceleration module:

Angle alpha between the total acceleration vector and normal acceleration (aka the angle between the total acceleration vector and the radius vector):

Please note that the total acceleration vector is not directed tangentially to the trajectory.

The tangential acceleration vector is directed along the tangent.

The direction of the total acceleration vector is determined by the vector sum of the normal and tangential acceleration vectors.

In this lesson we will look at important characteristic uneven movement - 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 next questions and their answers:

Question. Is acceleration free fall acceleration, according to the definition given 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.

As is known, motion in classical physics is described by Newton’s second law. Thanks to this law, the concept of body acceleration is introduced. In this article we will consider the basic concepts in physics that use acting force, speed and distance traveled by the body.

The concept of acceleration through Newton's second law

If for a while physical body mass m is acted upon by an external force F¯, then in the absence of other influences on it, we can write the following equality:

Here a¯ is called linear acceleration. As can be seen from the formula, it is directly proportional to the external force F¯, since the mass of the body can be considered a constant value at speeds much lower than the speed of propagation electromagnetic waves. In addition, the vector a¯ coincides in direction with F¯.

The above expression allows us to write the first acceleration formula in physics:

a¯ = F¯/m or a = F/m

Here the second expression is written in scalar form.

Acceleration, speed and distance traveled

Another way to find linear acceleration a¯ is to study the process of body motion along a straight path. Such movement is usually described by such characteristics as speed, time and distance traveled. In this case, acceleration is understood as the rate of change of the velocity itself.

For the rectilinear movement of objects, the following formulas in scalar form are valid:

2) a cp = (v 2 -v 1)/(t 2 -t 1);

3) a cp = 2*S/t 2

The first expression is defined as the derivative of speed with respect to time.

The second formula allows you to calculate the average acceleration. Here we consider two states of a moving object: its speed at time v 1 of time t 1 and a similar value v 2 at time t 2 . Time t 1 and t 2 is counted from some initial event. Note that the average acceleration generally characterizes this value over the considered time interval. Inside it, the value of instantaneous acceleration can change and differ significantly from the average a cp.

The third acceleration formula in physics also makes it possible to determine a cp, but already through the traversed path S. The formula is valid if the body began to move from zero speed, that is, when t=0, v 0 =0. This type of motion is called uniformly accelerated. His a shining example is the fall of bodies in the gravitational field of our planet.

Uniform circular motion and acceleration

As stated, acceleration is a vector and by definition represents the change in speed per unit time. In the case of uniform motion around a circle, the velocity module does not change, but its vector constantly changes direction. This fact leads to the emergence of a specific type of acceleration, called centripetal. It is directed to the center of the circle along which the body moves, and is determined by the formula:

a c = v 2 /r, where r is the radius of the circle.

This acceleration formula in physics demonstrates that its value increases faster with increasing speed than with decreasing radius of curvature of the trajectory.

An example of a c is the movement of a car entering a turn.

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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