The body was constant and the body traveled the same paths for any equal periods of time.

Most movements, however, cannot be considered uniform. In some areas of the body the speed may be lower, in others it may be higher. For example, a train leaving a station begins to move faster and faster. Approaching the station, he, on the contrary, slows down.

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

If the speed of a body during uneven motion changes equally over any equal periods of time, then the motion 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 approximately 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 a speed of 9.8 m/s, after another second - 19.6 m/s, after another second - 29.4 m/s, etc.

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

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/s 2 and is called “meter per second squared”.

Acceleration characterizes the rate of change in speed. If, for example, the acceleration of a 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 do we calculate the acceleration with which bodies begin to move?

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, we need to divide 1.2 m/s by 2 s. We get 0.6 m/s2. This is the acceleration of the train.

So, to find the acceleration of a body starting uniformly accelerated motion, it is necessary to divide the speed acquired by the body by the time during which this speed was achieved:

Let us denote all quantities included in this expression, with Latin letters:
a - acceleration; V- acquired speed; t - time

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

This formula is valid for uniformly accelerated motion from the state peace, i.e. when the initial speed of the body is zero. The initial speed of the body is denoted by V 0 - Formula (2.1), therefore, is valid only under the condition that V 0 = 0.

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

In this form, the acceleration formula is used in cases where a body having a certain speed V 0 begins to move slower and slower until it finally stops ( v= 0). It is by this formula, for example, that we will calculate the acceleration when braking cars and other Vehicle. By time t we will understand the braking time.

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 vector size. Therefore, in the pictures it is depicted as an arrow.

If the speed of a body at uniform acceleration straight motion increases, then the acceleration is directed in the same direction as the speed (Fig. 4, a); if the speed of the body decreases during a given movement, then the acceleration is directed in the opposite direction (Fig. 4, b).


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

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

Experimental task . Using the 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 this depend on the angle of the ruler?

S.V. Gromov, N.A. Rodina, Physics 8th grade

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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.

When moving in a straight line, 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.

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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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.

1. 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
2. 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
3. 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
4. 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
5. 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., however, this value conditionally coincides with what speed, acceleration 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.

In this topic we will look at a very special type of irregular motion. Based on the opposition to uniform motion, uneven movement- this 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 interval.

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).