For example, a car that starts off moves faster as it increases its speed. At the starting point, the speed of the car is zero. Starting the movement, the car accelerates to a certain speed. If you need to slow down, the car will not be able to stop instantly, but for some time. That is, the speed of the car will tend to zero - the car will start to move slowly until it stops completely. But physics does not have the term "deceleration". If the body moves, decreasing speed, this process is also called acceleration, but with a "-" sign.

Average acceleration is the ratio of the change in speed to the time interval during which this change occurred. Calculate the average acceleration using the formula:

where is it . The direction of the acceleration vector is the same as the direction of the change in speed Δ = - 0

where 0 is the initial speed. At the point in time t1(see figure below) the body has 0 . At the point in time t2 body has speed. Based on the vector subtraction rule, we determine the vector of speed change Δ = - 0 . From here we calculate the acceleration:

.

In the SI system unit of acceleration is called 1 meter per second per second (or meter per second squared):

.

A meter per second squared is the acceleration of a point moving in a straight line, at which the speed of this point increases by 1 m / s in 1 s. In other words, acceleration determines the degree of change in the speed of a body in 1 s. For example, if the acceleration is 5 m / s 2, then the speed of the body increases by 5 m / s every second.

Instant acceleration of the body ( material point) in this moment time is a physical quantity, which is equal to the limit to which the average acceleration tends when the time interval tends to 0. In other words, this is the acceleration developed by the body in a very small period of time:

.

The acceleration has the same direction as the change in speed Δ in extremely small time intervals during which the speed changes. The acceleration vector can be set using projections on the corresponding coordinate axes in a given reference system (projections a X, a Y , a Z).

With accelerated rectilinear motion the speed of the body increases modulo, i.e. v 2 > v 1 , and the acceleration vector has the same direction as the velocity vector 2 .

If the modulo velocity of the body decreases (v 2< v 1), значит, у вектора ускорения направление противоположно направлению вектора скорости 2 . Другими словами, в таком случае наблюдаем deceleration(acceleration is negative, and< 0). На рисунке ниже изображено направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

If there is a movement along a curvilinear trajectory, then the modulus and direction of the velocity changes. This means that the acceleration vector is represented as 2 components.

Tangential (tangential) acceleration call that component of the acceleration vector, which is directed tangentially to the trajectory at a given point of the trajectory of motion. Tangential acceleration describes the degree of change in speed modulo when making a curvilinear motion.

At tangential acceleration vectorsτ (see figure above) the direction is the same as that of the linear velocity or opposite to it. Those. the vector of tangential acceleration is in the same axis as the tangent circle, which is the trajectory of the body.

Translational and rotational movements

Translational this movement is called solid body, at which any straight line drawn in this body moves, remaining parallel to its initial direction.

Translational motion should not be confused with rectilinear. During the translational motion of the body, the trajectories of its points can be any curved lines.

The rotational motion of a rigid body around a fixed axis is such a motion in which any two points belonging to the body (or invariably associated with it) remain motionless throughout the motion.

Speed is the ratio of the distance traveled to the time it took to travel the distance.
The speed is the same is the sum of initial velocity and acceleration multiplied by time.
Speed is the product of the angular velocity and the radius of the circle.

v=S/t
v=v 0 +a*t
v=ωR

Acceleration of a body in uniformly accelerated motion- a value equal to the ratio of the change in speed to the time interval during which this change occurred.

Tangential (tangential) acceleration is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.

Rice. 1.10. tangential acceleration.

The direction of the tangential acceleration vector τ (see Fig. 1.10) coincides with the direction of the linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis as the tangent circle, which is the trajectory of the body.

Normal acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. That is, the normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in the direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.

Full acceleration in curvilinear motion, it consists of tangential and normal accelerations along vector addition rule and is determined by the formula:

(according to the Pythagorean theorem for a rectangular rectangle).

The direction of full acceleration is also determined vector addition rule:

angular velocity is called a vector quantity equal to the first derivative of the angle of rotation of the body with respect to time:

v=ωR

angular acceleration is called a vector quantity equal to the first derivative of the angular velocity with respect to time:

Fig.3

When the body rotates around a fixed axis, the angular acceleration vector ε is directed along the axis of rotation towards the vector of the elementary increment of the angular velocity. With accelerated movement, the vector ε co-directed to the vector ω (Fig. 3), when slowed down, it is opposite to it (Fig. 4).

Fig.4

Tangential acceleration component a τ =dv/dt , v = ωR and

Normal component of acceleration

This means that the relationship between linear (path length s, traveled by a point along an arc of radius R, linear velocity v, tangential acceleration a τ, normal acceleration a n) and angular quantities (angle of rotation φ, angular velocity ω, angular acceleration ε) is expressed as follows formulas:

s = Rφ, v = Rω, and τ = R?, a n = ω 2 R.
In the case of equally variable motion of a point along a circle (ω=const)

ω = ω 0 ± ?t, φ = ω 0 t ± ?t 2 /2,
where ω 0 is the initial angular velocity.

When bodies move, their velocities usually change either in absolute value, or in direction, or simultaneously both in absolute value and in direction.

If you throw a stone at an angle to the horizon, then its speed will change both in magnitude and in direction.

The change in the speed of the body can occur both very quickly (movement of a bullet in the bore when fired from a rifle), and relatively slowly (movement of a train when it is sent). To be able to find the speed at any moment of time, it is necessary to enter a value that characterizes the rate of change of speed. This value is calledacceleration.

- this is the ratio of the change in speed to the period of time during which this change occurred. The average acceleration can be determined by the formula:

where - acceleration vector .

The direction of the acceleration vector coincides with the direction of the change in speed Δ = - 0 (here 0 is the initial speed, that is, the speed at which the body began to accelerate).

At time t1 (see Figure 1.8) the body has a speed of 0 . At time t2 the body has a speed. According to the vector subtraction rule, we find the vector of speed change Δ = - 0 . Then the acceleration can be defined as follows:

Rice. 1.8. Average acceleration.

in SI unit of acceleration is 1 meter per second per second (or meter per second squared), that is

A meter per second squared is equal to the acceleration of a point moving in a straight line, at which in one second the speed of this point increases by 1 m / s. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m / s 2, then this means that the speed of the body increases by 5 m / s every second.

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 considering very big body, look for 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- 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 travels the same path in any equal intervals of time.

But most movements are uneven. In some areas, the speed of the body is greater, in others less. The car starts moving faster and faster. and when it stops, it slows down.

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

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/s2 and is called "meter per second squared".

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. a); if the speed of the body during this movement decreases, 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 has 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 \(\Delta t \to 0 \) . With the definition of the derivative of a function in mind, instantaneous acceleration can be defined as the time derivative of velocity:

\(\vec a = \dfrac (d\vec v) (dt) \)

Tangential and normal acceleration

If we write the speed as \(\vec v = v\hat \tau \) , where \(\hat \tau \) is the unit vector of the tangent to the motion trajectory, 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; \(\hat n \) - vector of the perpendicular to the velocity.

In this way,

\(\vec a = \vec a_(\tau) + \vec a_n \),

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 tangentially to the trajectory of motion, then \(\hat n \) is the vector 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 the speed, while normal characterizes the rate of change in its direction.

Movement along a curvilinear trajectory at each moment of time can be represented as a rotation around the center of curvature of the trajectory with an 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 the body will change per unit of time if it constantly moves with such an 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

• square meter per second, m/s², SI derived unit
• centimeter per second squared, cm/s², CGS derived unit