Average and instant acceleration and speed

Acceleration characterizes the speed of a change in speed both in magnitude and direction. You can find the average acceleration to determine the average rate of change in body velocity over a certain period of time. You may not be familiar with the calculation of acceleration (such as these are not everyday tasks), but this article will tell you how to quickly find the average acceleration.

Method 1 Calculating Average Acceleration

1. 1 Definition of acceleration. Acceleration is the speed at which speed increases or decreases, or simply the speed at which speed changes over time. Acceleration is a vector quantity that has a direction (include it in the answer).
   • Usually, if the body accelerates when moving “to the right”, “up” or “forward”, then the acceleration has a positive (+) value.
   • If the body accelerates when moving "left", "down" or "back", then the acceleration has a negative (+) value.
2. 2 Write the definition of acceleration as a formula. As mentioned above, acceleration is the speed at which speed changes over time. There are two ways to write this definition in the form of a formula:
   • a_wed = Δv /_Δt (The delta symbol "Δ" means "change").
   • a_wed = (v_to - v_n) /_{(tto - tn)} where v_to - final speed, v_n - starting speed.
3. 3 Find the starting and ending speeds of the body. For example, a car starting to move (to the right) from the parking lot has an initial speed of 0 m / s and a final speed of 500 m / s.
   • The movement to the right is described by positive values, so we will not indicate the direction of movement further.
   • If the car starts moving forward and ends with moving backwards, the final speed is negative.
4. 4 Note the change in time. For example, a car may take 10 seconds to reach its final speed. In this case, t_to = 10 s, and t_n = 0 s
   • Make sure that the speed and time are given in the appropriate units. For example, if speed is given in km / h, then time should be measured in hours.
5. 5 Substitute the speed and time values ​​given to you in the formula for calculating the average acceleration. In our example:
   • a_wed = (500 m / s - 0 m / s) /_{(10s - 0s)}
   • a_wed = (500 m / s) /_(10s)
   • a_wed = 50 m / s / s, i.e. 50 m / s 2.
6. 6 Interpretation of the result. The average acceleration sets the average rate of change of speed over a certain period of time. In the above example, the car accelerated on average by 50 m / s for every second. Remember: the motion parameters can be different, but the average acceleration will be the same only if the change in speed and change in time do not change:
   • The car can start moving at a speed of 0 m / s and accelerate in 10 seconds to 500 m / s.
   • The car can start moving at a speed of 0 m / s and accelerate to 900 m / s, and then slow down to 500 m / s in 10 seconds.
   • The car can start moving at a speed of 0 m / s, stand still for 9 seconds, and then accelerate to 500 m / s in 1 second.

Method 2 Positive and Negative Acceleration

1. 1 Determination of positive and negative speed. Speed ​​has a direction (since it is a vector quantity), but to indicate it, for example, as “up” or “north”, is very tiring. Instead, most tasks assume that the body moves along a straight line. When moving in one direction, the speed of the body is positive, and when moving in the opposite direction, the speed of the body is negative.
   • For example, a blue train moves east at a speed of 500 m / s. The red train moves west at the same speed, but since it moves in the opposite direction, its speed is written like this: -500 m / s.
2. 2 Use the definition of acceleration to determine its sign (+ or -). Acceleration - the speed of a change in speed over time. If you don’t know which sign to write for the acceleration value, find the change in speed:
   • v_{the ultimate} - v_initial = + or -?
3. 3 Acceleration in different directions. For example, the blue train and the red train move in opposite directions at a speed of 5 m / s. Imagine this movement on a number line, the blue train moves at a speed of 5 m / s in the positive direction of the number line (i.e., to the right), and the red train moves at a speed of -5 m / s in the negative direction of the number line (i.e., left). If each train increases speed by 2 m / s (in the direction of its movement), then which sign has acceleration? Let's check:
   • The blue train moves in a positive direction, so its speed increases from 5 m / s to 7 m / s. The final speed is 7 - 5 = +2. Since the change in speed is positive, acceleration is positive.
   • The red train moves in a negative direction and increases speed from -5 m / s to -7 m / s. The final speed is -7 - (-5) = -7 + 5 = -2 m / s. Since the change in speed is negative, acceleration is also negative.
4. 4 Slowdown. For example, a plane flies at a speed of 500 km / h, and then slows down to 400 km / h. Although the plane moves in a positive direction, its acceleration is negative, as it slows down (i.e. reduces speed). This can be checked through calculations: 400 - 500 = -100, that is, the change in speed is negative, and therefore the acceleration is negative.
   • On the other hand, if the helicopter moves at a speed of -100 km / h and accelerates to -50 km / h, then its acceleration is positive, because the change in speed is positive: -50 - (-100) = 50 (although such a change in speed was not enough to change the direction of movement of the helicopter).

Acceleration and speed are vector quantities that are specified by both value and direction. Values ​​given only by a value are called scalar (for example, length).

Finding speed

Every student knows about this concept, starting from the elementary grades. All students are familiar with the following formula:

Here S is the path that a moving body has covered in time t. This expression allows us to calculate some average velocity v. Indeed, we don’t know how the body moved, on which part of the path it moved faster, and on which slower. Even the situation is not excluded that at some point on the path it was at rest for some time. The only thing that is known is the distance traveled and the corresponding time period.

In high school, speed, as a physical quantity, is seen in a new light. Students are offered the following definition:

To understand this expression, you need to know how the derivative of a function is calculated. In this case, it is S (t). Since the derivative characterizes the behavior of the curve at this particular point, the speed calculated by the formula above is called instantaneous.

If the mechanical motion is variable, then for its accurate description it is necessary to know not only speed, but also a quantity that shows how it changes over time. This is the acceleration, which is the time derivative of the speed. And that, in turn, is a time derivative of the path. The formula for instant acceleration is:

Due to this equality, it is possible to determine the change in v at any point on the trajectory.

By analogy with speed, the average acceleration is calculated by the following formula:

Here Δv is the change in the module of the velocity of the body over the period of time Δt. Obviously, during this period the body is able to both accelerate and slow down. The value of a, determined from the expression above, will show only on average the speed of the change in speed.

Constant Acceleration

A distinctive feature of this type of movement of bodies in space is the constancy of the quantity a, that is, a = const.

This motion is also called uniformly accelerated or equally slow depending on the mutual direction of the velocity and acceleration vectors. Below we consider such a movement using the example of the two most common trajectories: a straight line and a circle.

When moving in a straight line during uniformly accelerated motion, the instantaneous speed and acceleration, as well as the distance traveled, are related by the following equalities:

Here v₀ is the value of speed that the body possessed before the appearance of acceleration a. Note one caveat. For this type of movement, it makes no sense to talk about instant acceleration, because at any point on the trajectory it will be the same. In other words, its instantaneous and average values ​​will be equal to each other.

As for speed, the first expression allows you to determine it at any time. That is, it will be an instant indicator. To calculate the average speed, you must use the above expression, that is:

Here t₁ and t₂ - these are time points between which the average speed is calculated.

The plus sign in all formulas corresponds to accelerated movement. Accordingly, the minus sign is in slow motion.

In the study of circular motion with constant acceleration in physics, angular characteristics are used that are similar to the corresponding linear ones. These include the angle of rotation θ, angular velocity and acceleration (ω and α). These values ​​are related in equalities, similar to expressions of uniformly accelerated motion in a straight line, which are given below:

In this case, the angular characteristics are associated with linear as follows:

Here R is the radius of the circle.

The task of determining the average and instantaneous acceleration

It is known that the body moves along a complex path. Its instantaneous speed varies in time as follows:

What is the instantaneous acceleration of the body at time t = 3 (seconds)? Find the average acceleration over a period of two to four seconds.

It is not difficult to answer the first question of the problem if we calculate the derivative of the function v (t). We get:

To determine the average acceleration, you should use this expression:

a = ((10 - 3 * 4 + 4 3) - (10 - 3 * 2 + 2 3)) / 2 = 25 m / s 2.

From the calculations it follows that the average acceleration slightly exceeds instantaneous in the middle of the considered time period.

Average acceleration

Average acceleration> Is the ratio of the change in speed to the period of time during which this change has occurred. You can determine the average acceleration by the formula:

Fig. 1.8. Average acceleration.In SI acceleration unit Is 1 meter per second per second (or meter per second squared), i.e.

A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which in one second the speed of this point increases by 1 m / s. In other words, acceleration determines how much the body’s speed 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.

Instant acceleration

Instant acceleration of the body (material point) at a given moment of time is a physical quantity equal to the limit towards which the average acceleration tends when the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:

With accelerated rectilinear motion, the velocity of the body increases in absolute value, i.e.

and the direction of the acceleration vector coincides with the velocity vector

If the body velocity decreases in absolute value, i.e.

then the direction of the acceleration vector is opposite to the direction of the velocity vector In other words, in this case slowdown, while the acceleration will be negative (a

Fig. 1.9. Instant acceleration.

When moving along a curved path, not only the speed modulus, but also its direction changes. In this case, the acceleration vector is represented in the form of two components (see the next section).

Tangential acceleration

Tangential (tangent) acceleration Is a component of the acceleration vector directed along the tangent to the trajectory at a given point of the motion path. Tangential acceleration characterizes the change in velocity modulo with curvilinear motion.

Fig. 1.10. Tangential acceleration.

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

Normal acceleration

Normal acceleration Is a component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. 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 indicated by the letter Normal acceleration vector is directed along the radius of curvature of the trajectory.

Full acceleration

Full acceleration during curvilinear motion it is composed of tangential and normal accelerations according to the rule of addition of vectors and is determined by the formula:

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

The direction of full acceleration is also determined by the rule of addition of vectors: