Initial speed acceleration. Moving during uniformly accelerated motion. Full acceleration vector

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§ 5. Acceleration.
Uniformly accelerated linear motion

1. With uneven movement, the speed of a body changes over time. Let's consider the simplest case of uneven motion.

Movement in which the speed of a body changes by the same value over any equal intervals of time is called uniformly accelerated.

For example, if for every 2 s the speed of a body changed by 4 m/s, then the movement of the body is uniformly accelerated. The velocity module during such movement can either increase or decrease.

2. Let at the initial moment of time t 0 = 0 the speed of the body is v 0 . At some point in time t she became equal v. Then the change in speed over a period of time t – t 0 = t equals v– v 0, and per unit time - . This relationship is called acceleration. Acceleration characterizes the rate of change in speed.

The acceleration of a body during uniformly accelerated motion is a vector physical quantity equal to the ratio of the change in the speed of the body to the period of time during which this change occurred.

a = .

The SI unit of acceleration is meters per second squared (1 ):

[a] === 1 .

The unit of acceleration is taken to be the acceleration of such uniformly accelerated motion, at which the speed of the body is 1 s changes to 1 m/s.

3. Since acceleration is a vector quantity, it is necessary to find out how it is directed.

Let the car move in a straight line with an initial speed v 0 (speed at time t= 0) and speed v at some point in time t. The vehicle's speed modulus increases. In Figure 22, A depicts a vector of car speed. From the definition of acceleration, it follows that the acceleration vector is directed in the same direction as the vector difference v–v 0 . Therefore, in this case, the direction of the acceleration vector coincides with the direction of motion of the body (with the direction of the velocity vector).

Let now the vehicle speed modulus decrease (Fig. 22 b). In this case, the direction of the acceleration vector is opposite to the direction of motion of the body (the direction of the velocity vector).

4. By transforming the acceleration formula for uniformly accelerated straight motion, you can get a formula for finding the speed of a body at any time:

v = v 0 + at.

If starting speed body is zero, i.e. at the initial moment of time it was at rest, then this formula takes the form:

v = at.

5. When calculating speed or acceleration, formulas are used that include not vectors, but projections of these quantities onto the coordinate axis. Since the projection of the sum of vectors is equal to the sum of their projections, the formula for the projection of velocity onto the axis X has the form:

v x = v 0x + a x t,

Where v x- projection of speed at a moment in time t, v 0x- projection of initial speed, a x- acceleration projection.

When solving problems, it is necessary to take into account the signs of projections. So, in the case shown in Figure 22, A, projections of velocities and acceleration onto the axis X positive; The speed modulus increases over time. In the case shown in Figure 22, b, projections onto the axis X velocities are positive, and the projection of acceleration is negative; the speed modulus decreases over time.

6. Example of problem solution

The vehicle speed during braking decreased from 23 to 15 m/s. What is the acceleration of the body if the braking lasts 5 s?

Given:

Solution

v 0 = 23 m/s

v= 15 m/s

t= 5 s

The car moves uniformly accelerated and in a straight line; its velocity modulus decreases.

We connect the reference system with the Earth, the axis X Let's direct it in the direction of the car's movement (Fig. 23), and take the beginning of braking as the beginning of the time count.

Let's write down the formula for finding the speed for uniformly accelerated rectilinear motion:

v = v 0 + at.

In projections onto the axis X we get

v x = v 0x + a x t.

Considering that the projection of the body’s acceleration onto the axis X is negative, and the projections of velocities on this axis are positive, we write: v = v 0 – at.

Where:

a = ;

a== 1.6 m/s 2 .

Answer: a= 1.6 m/s 2.

Self-test questions

1. What kind of motion is called uniformly accelerated?

2. What is the acceleration of uniformly accelerated motion called?

3. What formula is used to calculate acceleration during uniformly accelerated motion?

4. What is the SI unit of acceleration?

5. What formula is used to calculate the speed of a body in uniformly accelerated linear motion?

6. What is the sign of the acceleration projection onto the axis X in relation to the projection of the body’s speed onto the same axis, if the module of its speed increases; is it decreasing?

Task 5

1. What is the acceleration of the car if, 2 minutes after it started moving from rest, it acquired a speed of 72 km/h?

2. A train whose initial speed is 36 km/h accelerates with an acceleration of 0.5 m/s 2 . What speed will the train acquire in 20 s?

3. A car moving at a speed of 54 km/h stops at a traffic light for 15 s. What is the acceleration of the car?

4. What speed will the cyclist acquire 5 s after the start of braking, if his initial speed is 10 m/s and the acceleration during braking is 1.2 m/s 2?

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 occurs only when the speed of a body changes. If the speed of a body increases or decreases by a certain constant amount, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m/s2) 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 average acceleration over two speeds

Formula for calculating average acceleration. The average acceleration of a body is calculated from its initial and final speeds (speed is the speed of movement in a certain direction) and the time it takes the body to reach its final speed. Formula for calculating acceleration: a = Δv / Δt, where a is 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 = v k - v n And Δt = t k - t n, Where v to– final speed, v n- starting speed, t to– final time, t n– initial time.

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

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 interval (time change). You will get the average acceleration over a given period of time.
- If the final speed is less than the initial speed, 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 = Δv / Δt = (v k - v n)/(t k - t n)
  - Write the variables: v to= 46.1 m/s, v n= 18.5 m/s, t to= 2.47 s, t n= 0 s.
  - Calculation: a= (46.1 - 18.5)/2.47 = 11.17 m/s 2 .
- Example 2: A motorcycle starts braking at a speed of 22.4 m/s and stops after 2.55 s. Find the average acceleration.
  - Write the formula: a = Δv / Δt = (v k - v n)/(t k - t n)
  - Write the variables: v to= 0 m/s, v n= 22.4 m/s, t to= 2.55 s, t n= 0 s.
  - Calculation: A= (0 - 22.4)/2.55 = -8.78 m/s 2 .
Calculation of acceleration by force
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. This acceleration depends on the net 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 cut– resultant force acting on the body, m- body mass, a– acceleration of the body.
  - When working with this formula, use metric units, which measure mass in kilograms (kg), force in newtons (N), and acceleration in meters per second per second (m/s2).
2. Find the mass of the body. To do this, place the body on the scale and find its mass in grams. If you are considering 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 acceleration using the above formula, you need to convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.
3. Find the net force acting on the body. The resulting force is not balanced by other forces. If two differently 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 larger 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 action of this force.
  
  Rearrange 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 acceleration, divide the force by the mass of the accelerating body.
  - Force is directly proportional to acceleration, that is, the greater the force acting on a body, the faster it accelerates.
  - Mass is inversely proportional to acceleration, that is, the greater the mass of a body, the slower it accelerates.
4. Calculate the acceleration using the resulting formula. Acceleration is equal to the quotient of the resulting force acting on the body divided by its mass. Substitute the values given to you into this formula to calculate the acceleration of the body.
  - For example: a force equal to 10 N acts on a body weighing 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 positive value, if it is directed upward or to the right; acceleration is negative if it is directed downward or to the left. Check 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 . The wind blowing in westward, acts on the boat with a force of 100 N. Find the acceleration of the boat in the north direction.
3. Solution: Since the force is perpendicular to the direction of movement, it does not affect the movement in that direction. Therefore, the acceleration of the boat in the north direction will not change and will be equal to 2 m/s 2.
Resultant force. If several forces act on a body at once, find the resulting force, and then proceed to calculate the acceleration. Consider the following problem (in two-dimensional space):
- Vladimir pulls (on the right) a container of mass 400 kg 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 conditions of this problem are 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.

Acceleration is a familiar word. For non-engineers, it most often comes across in news articles and releases. Accelerating development, cooperation, others social processes. The original meaning of this word is associated with physical phenomena. How to find the acceleration of a moving body, or acceleration, as an indicator of the power of a car? Could it have other meanings?

What happens between 0 and 100 (term definition)

An indicator of a car's power is considered to be the time it takes to accelerate from zero to hundreds. What happens in between? Let's look at our Lada Vesta with its stated 11 seconds.

One of the formulas for finding acceleration is written like this:

a = (V 2 - V 1) / t

In our case:

a - acceleration, m/s∙s

V1 - initial speed, m/s;

V2 - final speed, m/s;

Let's bring the data into the SI system, namely, km/h will be converted to m/s:

100 km/h = 100000 m / 3600 s = 27.28 m/s.

Now you can find the acceleration of the "Kalina":

a = (27.28 - 0) / 11 = 2.53 m/s∙s

What do these numbers mean? An acceleration of 2.53 meters per second per second means that for every second the speed of the “car” increases by 2.53 m/s.

When starting from a place (from scratch):

in the first second the car will accelerate to a speed of 2.53 m/s;
for the second - up to 5.06 m/s;
by the end of the third second the speed will be 7.59 m/s, etc.

Thus, we can summarize: acceleration is the increase in the speed of a point per unit time.

Newton's second law, it's not difficult

So, the acceleration value has been calculated. It's time to ask where this acceleration comes from, what is its primary source. There is only one answer - strength. It is the force with which the wheels push the car forward that causes its acceleration. And how to find acceleration if the magnitude of this force is known? The relationship between these two quantities and the mass of a material point was established by Isaac Newton (this did not happen on the day when an apple fell on his head, then he discovered another physical law).

And this law is written like this:

F = m ∙ a, where

F - force, N;

m - mass, kg;

a - acceleration, m/s∙s.

In relation to a product of the Russian automobile industry, it is possible to calculate the force with which the wheels push the car forward.

F = m ∙ a = 1585 kg ∙ 2.53 m/s∙s = 4010 N

or 4010 / 9.8 = 409 kg∙s

This means that if you do not release the gas pedal, the car will accelerate until it reaches the speed of sound? Of course not. Already when it reaches a speed of 70 km/h (19.44 m/s), the frontal air resistance reaches 2000 N.

How to find the acceleration at the moment when the Lada “flies” at such a speed?

a = F / m = (F wheels - F resistance) / m = (4010 - 2000) / 1585 = 1.27 m/s∙s

As you can see, the formula allows you to find both acceleration, knowing the force with which the engines act on the mechanism (other forces: wind, water flow, weight, etc.), and vice versa.

Why is it necessary to know acceleration?

First of all, in order to calculate the speed of any material body at the moment of interest, as well as its location.

Suppose that our Lada Vesta accelerates on the Moon, where there is no frontal air resistance due to the lack of it, then its acceleration at some stage will be stable. In this case, we will determine the speed of the car 5 seconds after the start.

V = V 0 + a ∙ t = 0 + 2.53 ∙ 5 = 12.65 m/s

or 12.62 ∙ 3600 / 1000 = 45.54 km/h

V 0 - initial speed of the point.

And at what distance from the start will our lunar vehicle be at this moment? The easiest way to do this is to use universal formula coordinate definitions:

x = x 0 + V 0 t + (at 2) / 2

x = 0 + 0 ∙ 5 + (2.53 ∙ 5 2) / 2 = 31.63 m

x 0 - initial coordinate of the point.

This is exactly the distance that “Vesta” will have time to move away from the starting line in 5 seconds.

But in reality, in order to find the speed and acceleration of a point at a given point in time, in reality it is necessary to take into account and calculate many other factors. Of course, if the Lada Vesta gets to the moon, it won’t be soon; its acceleration, in addition to the power of the new injection engine, is affected not only by air resistance.

At different engine speeds, it produces different forces, without taking into account the number of the gear engaged, the coefficient of adhesion of the wheels to the road, the slope of this very road, wind speed and much more.

What other accelerations are there?

Strength does more than just force the body to move forward in a straight line. For example, the gravitational force of the Earth causes the Moon to constantly bend its flight path in such a way that it always circles around us. Is there a force acting on the Moon in this case? Yes, this is the same force that was discovered by Newton with the help of an apple - the force of attraction.

And the acceleration that it gives to our natural satellite is called centripetal. How to find the acceleration of the Moon as it moves in orbit?

a c = V 2 / R = 4π 2 R / T 2, where

a c - centripetal acceleration, m/s∙s;

V is the speed of the Moon’s orbit, m/s;

R - orbital radius, m;

T is the period of revolution of the Moon around the Earth, s.

a c = 4 π 2 384 399 000 / 2360591 2 = 0.002723331 m/s∙s

How do the speedometer readings change when starting to move and when the car brakes?
Which physical quantity characterizes the change in speed?

When bodies move, their speeds usually change either in magnitude or in direction, or at the same time both in magnitude and in direction.

The speed of a puck sliding on ice decreases over time until it stops completely. If you pick up a stone and unclench your fingers, then as the stone falls, its speed gradually increases. The speed of any point on the circle of the grinding wheel, with a constant number of revolutions per unit time, changes only in direction, remaining constant in magnitude (Figure 1.26). If you throw a stone at an angle to the horizon, then its speed will change both in magnitude and direction.

A change in the speed of a body can occur either very quickly (the movement of a bullet in the barrel when fired from a rifle) or relatively slowly (the movement of a train when it departs).

A physical quantity characterizing the rate of change of speed is called acceleration.

Let us consider the case of curvilinear and uneven motion of a point. In this case, its speed changes over time both in magnitude and direction. Let at some moment of time t the point occupy a position M and have a speed (Fig. 1.27). After a period of time Δt, the point will take position M 1 and will have a speed of 1. The change in speed over time Δt 1 is equal to Δ 1 = 1 - . Subtracting a vector can be done by adding 1 vector (-) to the vector:

Δ 1 = 1 - = 1 + (-).

According to the rule of vector addition, the speed change vector Δ 1 is directed from the beginning of vector 1 to the end of vector (-), as shown in Figure 1.28.

Dividing the vector Δ 1 by the time interval Δt 1 we obtain a vector directed in the same way as the vector of change in speed Δ 1 . This vector is called the average acceleration of a point over a period of time Δt 1. Denoting it by ср1, we write:

By analogy with the definition of instantaneous speed, we define instantaneous acceleration. To do this, we now find the average accelerations of the point over smaller and smaller periods of time:

As the time period Δt decreases, the vector Δ decreases in magnitude and changes in direction (Fig. 1.29). Accordingly, the average accelerations also change in magnitude and direction. But as the time interval Δt tends to zero, the ratio of the change in speed to the change in time tends to a certain vector as its limiting value. In mechanics, this quantity is called the acceleration of a point at a given moment in time or simply acceleration and is denoted .

The acceleration of a point is the limit of the ratio of the change in speed Δ to the time period Δt during which this change occurred, as Δt tends to zero.

Acceleration is directed in the same way as the vector of change in speed Δ is directed as the time interval Δt tends to zero. Unlike the direction of velocity, the direction of the acceleration vector cannot be determined by knowing the trajectory of the point and the direction of movement of the point along the trajectory. In the future on simple examples we will see how we can determine the direction of acceleration of a point during rectilinear and curvilinear motion.

In the general case, acceleration is directed at an angle to the velocity vector (Fig. 1.30). Total acceleration characterizes the change in speed both in magnitude and direction. Often the total acceleration is considered equal to the vector sum of two accelerations - tangential (k) and centripetal (cs). Tangential acceleration k characterizes the change in speed in magnitude and is directed tangentially to the trajectory of motion. Centripetal acceleration cs characterizes the change in speed in the direction and perpendicular to the tangent, i.e., directed towards the center of curvature of the trajectory at a given point. In the future, we will consider two special cases: a point moves in a straight line and the speed changes only in absolute value; the point moves uniformly around the circle and the speed changes only in direction.

Unit of acceleration.

The movement of a point can occur with both variable and constant acceleration. If the acceleration of a point is constant, then the ratio of the change in speed to the period of time during which this change occurred will be the same for any time interval. Therefore, denoting by Δt some arbitrary period of time, and by Δ the change in speed over this period, we can write:

Since the time period Δt is a positive quantity, it follows from this formula that if the acceleration of a point does not change over time, then it is directed in the same way as the velocity change vector. Thus, if acceleration is constant, then it can be interpreted as the change in speed per unit time. This allows you to set the units of the acceleration modulus and its projections.

Let's write the expression for the acceleration module:

It follows that:
acceleration module numerically equal to one, if per unit time the module of the velocity change vector changes by one.
If time is measured in seconds and speed is measured in meters per second, then the unit of acceleration is m/s 2 (meter per second squared).

Let's take a closer look at what acceleration is in physics? This is a message to the body of additional speed per unit of time. In the International System of Units (SI), the unit of acceleration is considered to be the number of meters traveled per second (m/s). For the extra-system unit of measurement Gal (Gal), which is used in gravimetry, the acceleration is 1 cm/s 2 .

Types of accelerations

What is acceleration in formulas. The type of acceleration depends on the vector of motion of the body. In physics, this can be motion in a straight line, along a curved line, or in a circle.

If an object moves in a straight line, the movement will be uniformly accelerated, and linear accelerations will begin to act on it. The formula for calculating it (see formula 1 in Fig.): a=dv/dt
If we are talking about the movement of a body in a circle, then the acceleration will consist of two parts (a=a t +a n): tangential and normal acceleration. Both of them are characterized by the speed of movement of the object. Tangential - changing the speed modulo. Its direction is tangential to the trajectory. This acceleration is calculated by the formula (see formula 2 in Fig.): a t =d|v|/dt
If the speed of an object moving around a circle is constant, the acceleration is called centripetal or normal. The vector of such acceleration is constantly directed towards the center of the circle, and the modulus value is equal to (see formula 3 in Fig): |a(vector)|=w 2 r=V 2 /r
When the speed of a body around a circle is different, angular acceleration occurs. It shows how the angular velocity has changed per unit time and is equal to (see formula 4 in the figure): E(vector)=dw(vector)/dt
Physics also considers options when a body moves in a circle, but at the same time approaches or moves away from the center. In this case, the object is affected by Coriolis accelerations. When the body moves along a curved line, its acceleration vector will be calculated by the formula (see formula 5 in Fig): a (vector)=a T T+a n n(vector)+a b b(vector) =dv/dtT+v 2 /Rn(vector)+a b b(vector), in which:

v - speed
T (vector) - unit vector tangent to the trajectory, running along the velocity (tangent unit vector)
n (vector) - unit vector of the main normal relative to the trajectory, which is defined as a unit vector in the direction dT (vector)/dl
b (vector) - unit of binormal relative to the trajectory
R - radius of curvature of the trajectory

In this case, the binormal acceleration a b b(vector) is always equal to zero. Therefore, the final formula looks like this (see formula 6 in Fig.): a (vector)=a T T+a n n(vector)+a b b(vector)=dv/dtT+v 2 /Rn(vector)

What is the acceleration of gravity?

Acceleration free fall(denoted by the letter g) is the acceleration that is imparted to an object in a vacuum by gravity. According to Newton's second law, this acceleration is equal to the force of gravity acting on an object of unit mass.

On the surface of our planet, the value of g is usually called 9.80665 or 10 m/s². To calculate the actual g on the Earth's surface, you will need to take into account some factors. For example, latitude and time of day. So the value of true g can be from 9.780 m/s² to 9.832 m/s² at the poles. To calculate it, an empirical formula is used (see formula 7 in Fig.), in which φ is the latitude of the area, and h is the distance above sea level, expressed in meters.

Formula for calculating g

The fact is that such free fall acceleration consists of gravitational and centrifugal acceleration. The approximate value of the gravitational value can be calculated by imagining the Earth as a homogeneous ball with mass M, and calculating the acceleration over its radius R (formula 8 in Fig, where G is the gravitational constant with a value of 6.6742·10 −11 m³s −2 kg −1) .

If we use this formula to calculate gravitational acceleration on the surface of our planet (mass M = 5.9736 10 24 kg, radius R = 6.371 10 6 m), we get formula 9 in Fig. given value conditionally coincides with what speed and acceleration are in a specific place. The discrepancies are explained by several factors:

Centrifugal acceleration taking place in the reference frame of the planet's rotation
Because planet Earth is not spherical
Because our planet is heterogeneous

Instruments for measuring acceleration

Acceleration is usually measured with an accelerometer. But it does not calculate the acceleration itself, but the ground reaction force that occurs during accelerated movement. The same resistance forces appear in the gravitational field, so gravity can also be measured with an accelerometer.

There is another device for measuring acceleration - an accelerograph. It calculates and graphically records the acceleration values of translational and rotational motion.