Acceleration characterizes the rate of change in the speed of a moving body. If the speed of a body remains constant, then it does not accelerate. Acceleration takes place only when the speed of the body changes. If the speed of a body increases or decreases by some constant value, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m/s 2) and is calculated from the values ​​of two speeds and time, or from the value of the force applied to the body.

Steps

Calculation of the average acceleration over two speeds

Formula for calculating the average acceleration. The average acceleration of a body is calculated from its initial and final velocities (speed is the speed of movement in a certain direction) and the time it takes the body to reach the final speed. Formula for calculating acceleration: a = ∆v / ∆t, where a is the acceleration, Δv is the change in speed, Δt is the time required to reach the final speed.

Definition of variables. You can calculate Δv and Δt in the following way: Δv \u003d v to - v n and Δt \u003d t to - t n, where v to- final speed v n- starting speed, t to- end time t n- start time.

• Since acceleration has a direction, always subtract the initial velocity from the final velocity; otherwise, the direction of the calculated acceleration will be wrong.
• If the initial time is not given in the problem, then it is assumed that t n = 0.

1. Find the acceleration using the formula. First, write the formula and the variables given to you. Formula: . Subtract the initial speed from the final speed, and then divide the result by the time span (change in time). You will get the average acceleration for a given period of time.

   • If the final speed is less than the initial one, then the acceleration has a negative value, that is, the body slows down.
   • Example 1: A car accelerates from 18.5 m/s to 46.1 m/s in 2.47 s. Find the average acceleration.
     • Write the formula: a \u003d Δv / Δt \u003d (v to - v n) / (t to - t n)
     • Write variables: v to= 46.1 m/s, v n= 18.5 m/s, t to= 2.47 s, t n= 0 s.
     • Calculation: a\u003d (46.1 - 18.5) / 2.47 \u003d 11.17 m / s 2.
   • Example 2: A motorcycle starts braking at 22.4 m/s and stops after 2.55 seconds. Find the average acceleration.
     • Write the formula: a \u003d Δv / Δt \u003d (v to - v n) / (t to - t n)
     • Write variables: v to= 0 m/s, v n= 22.4 m/s, t to= 2.55 s, t n= 0 s.
     • Calculation: a\u003d (0 - 22.4) / 2.55 \u003d -8.78 m / s 2.

   Force Acceleration Calculation

   1. Newton's second law. According to Newton's second law, a body will accelerate if the forces acting on it do not balance each other. Such acceleration depends on the resultant force acting on the body. Using Newton's second law, you can find the acceleration of a body if you know its mass and the force acting on that body.

      • Newton's second law is described by the formula: F res = m x a, where F res is the resultant force acting on the body, m- body mass, a is the acceleration of the body.
      • When working with this formula, use the units of the metric system, in which mass is measured in kilograms (kg), force in newtons (N), and acceleration in meters per second per second (m/s 2).

   2. Find the mass of the body. To do this, put the body on the scales and find its mass in grams. If you are looking at a very large body, look up its mass in reference books or on the Internet. The mass of large bodies is measured in kilograms.

      • To calculate the acceleration using the above formula, you must convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.

   3. Find the resultant force acting on the body. The resulting force is not balanced by other forces. If two oppositely directed forces act on a body, and one of them is greater than the other, then the direction of the resulting force coincides with the direction of the greater force. Acceleration occurs when a force acts on a body that is not balanced by other forces and which leads to a change in the speed of the body in the direction of this force.

      Transform the formula F = ma to calculate the acceleration. To do this, divide both sides of this formula by m (mass) and get: a = F / m. Thus, to find the acceleration, divide the force by the mass of the accelerating body.

      • The force is directly proportional to the acceleration, that is, the greater the force acting on the body, the faster it accelerates.
      • Mass is inversely proportional to acceleration, that is, the greater the mass of the body, the slower it accelerates.

   4. Calculate the acceleration using the resulting formula. Acceleration is equal to the quotient of the resultant force acting on the body divided by its mass. Substitute the values ​​given to you into this formula to calculate the body's acceleration.

      • For example: a force equal to 10 N acts on a body of mass 2 kg. Find the acceleration of the body.
      • a = F/m = 10/2 = 5 m/s 2

   Testing your knowledge

   1. direction of acceleration. The scientific concept of acceleration does not always coincide with the use of this quantity in everyday life. Remember that acceleration has a direction; acceleration has a positive value if it is directed upwards or to the right; acceleration has a negative value if it is directed downwards or to the left. Check the correctness of your solution based on the following table:

   2. Example: a toy boat with a mass of 10 kg is moving north with an acceleration of 2 m/s 2 . A wind blowing in a westerly direction acts on a boat with a force of 100 N. Find the acceleration of the boat in a northerly direction.
   3. Solution: Since the force is perpendicular to the direction of motion, it does not affect the motion in that direction. Therefore, the acceleration of the boat in the northern direction will not change and will be equal to 2 m / s 2.

2. resultant force. If several forces act on the body at once, find the resulting force, and then proceed to calculate the acceleration. Consider the following problem (in two dimensions):

   • Vladimir pulls (on the right) a 400 kg container with a force of 150 N. Dmitry pushes (on the left) a container with a force of 200 N. The wind blows from right to left and acts on the container with a force of 10 N. Find the acceleration of the container.
   • Solution: The condition of this problem is designed to confuse you. In fact, everything is very simple. Draw a diagram of the direction of forces, so you will see that a force of 150 N is directed to the right, a force of 200 N is also directed to the right, but a force of 10 N is directed to the left. Thus, the resulting force is: 150 + 200 - 10 = 340 N. The acceleration is: a = F / m = 340/400 = 0.85 m / s 2.

In the 7th grade physics course, you studied the simplest kind of motion - uniform motion in a straight line. With such a movement, the speed of the body was constant and the body covered the same paths for any equal time intervals.

Most movements, however, cannot be considered uniform. In some parts of the body they may have a lower speed, in others - a greater one. For example, a train leaving a station starts moving faster and faster. Approaching the station, he, on the contrary, slows down his movement.

Let's do an experiment. We install a dropper on the cart, from which drops of a colored liquid fall at regular intervals. Let's place this cart on an inclined board and let it go. We will see that the distance between the traces left by the drops will become larger and larger as the cart moves down (Fig. 3). This means that the cart travels unequal distances in equal time intervals. The speed of the cart increases. Moreover, as can be proved, for the same time intervals, the speed of a cart moving down an inclined board increases all the time by the same amount.

If the speed of the body during uneven movement for any equal intervals of time changes in the same way, then the movement is called uniformly accelerated.

So, for example, experiments have established that the speed of any freely falling body (in the absence of air resistance) increases by about 9.8 m / s every second, i.e. if at first the body was at rest, then a second after the start of the fall it will have the speed is 9.8 m / s, after another second - 19.6 m / s, after another second - 29.4 m / s, etc.

A physical quantity showing how much the speed of a body changes for every second of uniformly accelerated motion is called acceleration.

a - acceleration.

The unit of acceleration in SI is such an 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 / s 2 and is called "meter per second squared."

Acceleration characterizes the rate of change of speed. If, for example, the acceleration of the body is 10 m / s 2, then this means that for every second the speed of the body changes by 10 m / s, i.e. 10 times faster than with an acceleration of 1 m / s 2.

Examples of accelerations encountered in our lives can be found in Table 1.


How is the acceleration with which the bodies begin to move calculated?

Let, for example, it is known that the speed of an electric train leaving the station increases by 1.2 m/s in 2 s. Then, in order to find out how much it increases in 1 s, you need to divide 1.2 m / s by 2 s. We will get 0.6 m / s 2. This is the acceleration of the train.

So, in order to find the acceleration of a body that starts uniformly accelerated motion, it is necessary to divide the speed acquired by the body by the time during which this speed was reached:

Let us denote all the quantities included in this expression in Latin letters:

a - acceleration; v - acquired speed; t - time.

Then the formula for determining the acceleration can be written as follows:

This formula is valid for uniformly accelerated motion from a state of rest, i.e., when the initial velocity of the body is zero. The initial velocity of the body is denoted by Formula (2.1), thus, it is valid to pour, provided that v 0 = 0.

If zero is not the initial, but the final speed (which is denoted simply by the letter v), then the acceleration formula takes the form:

In this form, the acceleration formula is used in cases where a body with a certain speed v 0 starts moving slower and slower until it finally stops (v \u003d 0). It is by this formula, for example, that we will calculate the acceleration when braking cars and other vehicles. By the time t we mean the deceleration time.

Like speed, body acceleration is characterized not only by a numerical value, but also by direction. This means that acceleration is also a vector quantity. Therefore, in the figures it is depicted as an arrow.

If the speed of the body during uniformly accelerated rectilinear motion increases, then the acceleration is directed in the same direction as the speed (Fig. 4, a); if the speed of the body during this movement decreases, then the acceleration is directed in the opposite direction (Fig. 4, b).

In uniform rectilinear motion, the speed of the body does not change. Therefore, there is no acceleration during such a motion (a = 0) and cannot be shown in the figures.

1. What movement is called uniformly accelerated? 2. What is acceleration? 3. What characterizes acceleration? 4. In what cases is the acceleration equal to zero? 5. What is the formula for the acceleration of a body during uniformly accelerated motion from a state of rest? 6. What is the formula for the acceleration of the body when the speed decreases to zero? 7. What is the direction of acceleration in uniformly accelerated rectilinear motion?

Experimental task. Using a ruler as an inclined plane, place a coin on its top edge and release. Will the coin move? If so, how - uniformly or uniformly accelerated? How does it depend on the angle of the ruler?

Content:

Acceleration characterizes the rate of change in the speed of a moving body. If the speed of a body remains constant, then it does not accelerate. Acceleration takes place only when the speed of the body changes. If the speed of a body increases or decreases by some constant value, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m/s 2) and is calculated from the values ​​of two speeds and time, or from the value of the force applied to the body.

Steps

1 Calculation of the average acceleration over two speeds

1. 1 Formula for calculating the average acceleration. The average acceleration of a body is calculated from its initial and final velocities (speed is the speed of movement in a certain direction) and the time it takes the body to reach the final speed. Formula for calculating acceleration: a = ∆v / ∆t, where a is the acceleration, Δv is the change in speed, Δt is the time required to reach the final speed.
   • The units of acceleration are meters per second per second, that is, m/s 2 .
   • Acceleration is a vector quantity, that is, it is given both by value and direction. Value is a numerical characteristic of acceleration, and direction is the direction of movement of the body. If the body slows down, then the acceleration will be negative.
2. 2 Definition of variables. You can calculate Δv and Δt in the following way: Δv \u003d v to - v n and Δt \u003d t to - t n, where v to- final speed v n- starting speed, t to- end time t n- start time.
   • Since acceleration has a direction, always subtract the initial velocity from the final velocity; otherwise, the direction of the calculated acceleration will be wrong.
   • If the initial time is not given in the problem, then it is assumed that t n = 0.
3. 3 Find the acceleration using the formula. First, write the formula and the variables given to you. Formula: . Subtract the initial speed from the final speed, and then divide the result by the time span (change in time). You will get the average acceleration for a given period of time.
   • If the final speed is less than the initial one, then the acceleration has a negative value, that is, the body slows down.
   • Example 1: A car accelerates from 18.5 m/s to 46.1 m/s in 2.47 s. Find the average acceleration.
     • Write the formula: a \u003d Δv / Δt \u003d (v to - v n) / (t to - t n)
     • Write variables: v to= 46.1 m/s, v n= 18.5 m/s, t to= 2.47 s, t n= 0 s.
     • Calculation: a\u003d (46.1 - 18.5) / 2.47 \u003d 11.17 m / s 2.
   • Example 2: A motorcycle starts braking at 22.4 m/s and stops after 2.55 seconds. Find the average acceleration.
     • Write the formula: a \u003d Δv / Δt \u003d (v to - v n) / (t to - t n)
     • Write variables: v to= 0 m/s, v n= 22.4 m/s, t to= 2.55 s, t n= 0 s.
     • Calculation: a\u003d (0 - 22.4) / 2.55 \u003d -8.78 m / s 2.

2 Calculation of acceleration by force

1. 1 Newton's second law. According to Newton's second law, a body will accelerate if the forces acting on it do not balance each other. Such acceleration depends on the resultant force acting on the body. Using Newton's second law, you can find the acceleration of a body if you know its mass and the force acting on that body.
   • Newton's second law is described by the formula: F res = m x a, where F res is the resultant force acting on the body, m- body mass, a is the acceleration of the body.
   • When working with this formula, use the units of the metric system, in which mass is measured in kilograms (kg), force in newtons (N), and acceleration in meters per second per second (m/s 2).
2. 2 Find the mass of the body. To do this, put the body on the scales and find its mass in grams. If you are looking at a very large body, look up its mass in reference books or on the Internet. The mass of large bodies is measured in kilograms.
   • To calculate the acceleration using the above formula, you must convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.
3. 3 Find the resultant force acting on the body. The resulting force is not balanced by other forces. If two oppositely directed forces act on a body, and one of them is greater than the other, then the direction of the resulting force coincides with the direction of the greater force. Acceleration occurs when a force acts on a body that is not balanced by other forces and which leads to a change in the speed of the body in the direction of this force.
   • For example, you and your brother are pulling a rope. You are pulling the rope with a force of 5 N and your brother is pulling the rope (in the opposite direction) with a force of 7 N. The net force is 2 N and is directed towards your brother.
   • Remember that 1 N \u003d 1 kg∙m / s 2.
4. 4 Transform the formula F = ma to calculate the acceleration. To do this, divide both sides of this formula by m (mass) and get: a = F / m. Thus, to find the acceleration, divide the force by the mass of the accelerating body.
   • The force is directly proportional to the acceleration, that is, the greater the force acting on the body, the faster it accelerates.
   • Mass is inversely proportional to acceleration, that is, the greater the mass of the body, the slower it accelerates.
5. 5 Calculate the acceleration using the resulting formula. Acceleration is equal to the quotient of the resultant force acting on the body divided by its mass. Substitute the values ​​given to you into this formula to calculate the body's acceleration.
   • For example: a force equal to 10 N acts on a body of mass 2 kg. Find the acceleration of the body.
   • a = F/m = 10/2 = 5 m/s 2

3 Testing your knowledge

1. 1 direction of acceleration. The scientific concept of acceleration does not always coincide with the use of this quantity in everyday life. Remember that acceleration has a direction; acceleration has a positive value if it is directed upwards or to the right; acceleration has a negative value if it is directed downwards or to the left. Check the correctness of your solution based on the following table:
2. 2 Direction of force. Remember that acceleration is always co-directional with the force acting on the body. Some tasks give data that is intended to mislead you.
   • Example: a toy boat with a mass of 10 kg is moving north with an acceleration of 2 m/s 2 . A wind blowing in a westerly direction acts on a boat with a force of 100 N. Find the acceleration of the boat in a northerly direction.
   • Solution: Since the force is perpendicular to the direction of motion, it does not affect the motion in that direction. Therefore, the acceleration of the boat in the northern direction will not change and will be equal to 2 m / s 2.
3. 3 resultant force. If several forces act on the body at once, find the resulting force, and then proceed to calculate the acceleration. Consider the following problem (in two dimensions):
   • Vladimir pulls (on the right) a 400 kg container with a force of 150 N. Dmitry pushes (on the left) a container with a force of 200 N. The wind blows from right to left and acts on the container with a force of 10 N. Find the acceleration of the container.
   • Solution: The condition of this problem is designed to confuse you. In fact, everything is very simple. Draw a diagram of the direction of forces, so you will see that a force of 150 N is directed to the right, a force of 200 N is also directed to the right, but a force of 10 N is directed to the left. Thus, the resulting force is: 150 + 200 - 10 = 340 N. The acceleration is: a = F / m = 340/400 = 0.85 m / s 2.

All tasks in which there is movement of objects, their movement or rotation, are somehow connected with speed.

This term characterizes the movement of an object in space over a certain period of time - the number of units of distance per unit of time. He is a frequent "guest" of both sections of mathematics and physics. The original body can change its location both uniformly and with acceleration. In the first case, the speed is static and does not change during the movement, in the second, on the contrary, it increases or decreases.

How to find speed - uniform motion

If the speed of the body remained unchanged from the beginning of the movement to the end of the path, then we are talking about moving with constant acceleration - uniform movement. It can be straight or curved. In the first case, the trajectory of the body is a straight line.

Then V=S/t, where:

• V is the desired speed,
• S - distance traveled (total path),
• t is the total time of movement.

How to find speed - acceleration is constant

If an object was moving with acceleration, then its speed changed as it moved. In this case, the expression will help to find the desired value:

V \u003d V (beginning) + at, where:

• V (beginning) - the initial speed of the object,
• a is the acceleration of the body,
• t is the total travel time.

How to find speed - uneven motion

In this case, there is a situation when the body passes different parts of the path in different times.
S(1) - for t(1),
S(2) - for t(2), etc.

On the first section, the movement took place at a “tempo” V(1), on the second - V(2), and so on.

To find out the speed of an object moving all the way (its average value), use the expression:

How to find speed - rotation of an object

In the case of rotation, we are talking about the angular velocity, which determines the angle through which the element rotates per unit of time. The desired value is denoted by the symbol ω (rad / s).

• ω = Δφ/Δt, where:

Δφ – passed angle (angle increment),
Δt - elapsed time (movement time - time increment).

• If the rotation is uniform, the desired value (ω) is associated with such a concept as the period of rotation - how long will it take for our object to make 1 complete revolution. In this case:

ω = 2π/T, where:
π is a constant ≈3.14,
T is the period.

Or ω = 2πn, where:
π is a constant ≈3.14,
n is the frequency of circulation.

• With the known linear speed of the object for each point on the path of motion and the radius of the circle along which it moves, the following expression is required to find the speed ω:

ω = V/R, where:
V is the numerical value of the vector quantity (linear velocity),
R is the radius of the body's trajectory.

How to find speed - approaching and moving away points

In such tasks, it would be appropriate to use the terms approach speed and distance speed.

If the objects are heading towards each other, then the speed of approach (retreat) will be as follows:
V (approach) = V(1) + V(2), where V(1) and V(2) are the velocities of the corresponding objects.

If one of the bodies catches up with the other, then V (closer) = V(1) - V(2), V(1) is greater than V(2).

How to find speed - movement on a body of water

If events unfold on the water, then the speed of the current (i.e., the movement of water relative to a fixed shore) is added to the object’s own speed (movement of the body relative to the water). How are these concepts related?

In the case of moving downstream, V=V(own) + V(tech).
If against the current - V \u003d V (own) - V (flow).

In this topic, we will consider a very special kind of non-uniform motion. Based on the opposition to uniform movement, uneven movement is movement at an unequal speed, along any trajectory. What is the characteristic of uniformly accelerated motion? This is an uneven movement, but which "equally accelerating". Acceleration is associated with an increase in speed. Remember the word "equal", we get an equal increase in speed. And how to understand "an equal increase in speed", how to evaluate the speed is equally increasing or not? To do this, we need to detect the time, estimate the speed through the same time interval. For example, a car starts moving, in the first two seconds it develops a speed of up to 10 m/s, in the next two seconds 20 m/s, after another two seconds it is already moving 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 each time the speed increases is called acceleration.

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

Usually, the movement with increasing speed is called accelerated movement. Movement with decreasing speed - slow motion. But physicists call any motion with a changing speed accelerated motion. Whether the car starts off (speed increases!), or slows down (speed decreases!), in any case, it moves with acceleration.

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

body acceleration

Acceleration characterizes the rate of change of speed. This is the number by which the speed changes every second. If the modulo acceleration of the body is large, this means that the body quickly picks up speed (when it accelerates) or quickly loses it (when decelerating). Acceleration- this 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 following 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 do we determine? We consider the speed difference in 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 task: 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, 11-7= 4, then 4/2=2. We divide the speed difference by the time interval.

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

The formula is not written in vector form, so we write the "+" sign when the body accelerates, the "-" sign - when it slows down.

Direction of the acceleration vector

The direction of the acceleration vector is shown in the figures

In this figure, the car is moving 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 speed, this means that the car is accelerating. The acceleration is positive.

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

In this picture, the car is moving in the positive direction on the Ox axis, the velocity vector is the same as the direction of movement (rightward), the acceleration is NOT the same as the direction of the speed, which means that the car is decelerating. The acceleration is negative.

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

Let's figure out why the acceleration is negative when braking. For example, in the first second, the ship dropped 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. That's where the negative acceleration value comes from.

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

Moving with uniformly accelerated motion

An additional formula called untimely

Formula in coordinates

Communication with medium speed

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

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

Path ratio

If the body moves uniformly accelerated, the initial speed is zero, then the paths traveled in successive equal time intervals are related as a 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 the 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 go towards each other: one - accelerated to the north, the other - slowly to the south. How are train accelerations directed?

Same to the north. Because the first train has the same acceleration in the direction of movement, and the second has the opposite movement (it slows down).