INSTANTANEOUS SPEED: DEFINITION, FORMULA, CALCULATION AND EXERCISES - PHYSICAL - 2024

The instantaneous velocity is defined as the instantaneous change of the time shift. It is a concept that adds great precision to the study of movement. And it is an advance with respect to the average speed, whose information is very general.

To get the instantaneous velocity, let's look at as small a time interval as possible. Differential calculus is the perfect tool to express this idea mathematically.

Instantaneous speed shows the speed of the mobile at each point of its journey. Source: Pixabay.

The starting point is the average speed:

This limit is known as a derivative. In the differential calculus notation we have:

As long as the motion is restricted to a straight line, the vector notation can be dispensed with.

Calculation of instantaneous velocity: geometric interpretation

The following figure shows the geometric interpretation of the derivative concept: it is the slope of the tangent line to the curve x (t) vs. t at each point.

The instantaneous velocity at P is numerically equal to the slope of the tangent line to the curve x vs. t at point P. Source: Source: す じ に く シ チ ュ ー.

You can imagine how to obtain the limit if point Q is approached little by little to point P. There will come a moment when both points are so close, that you will not be able to distinguish one from the other.

The line that joins them will then go from being secant (line that intersects at two points) to being tangent (line that touches the curve at only one point). Therefore, to find the instantaneous velocity of a moving particle we should have:

• The graph of the position of the particle as a function of time. Finding the slope of the tangent line to the curve at each instant of time, we have the instantaneous velocity at each point that the particle occupies.

O well:

• The position function of the particle x (t), which is derived to obtain the velocity function v (t), then this function is evaluated at each time t, at convenience. The position function is assumed to be differentiable.

Some special cases in calculating instantaneous velocity

-The slope of the tangent line to the curve at P is 0. A null slope means that the mobile is stopped and that its speed is of course 0.

-The slope of the tangent line to the curve at P is greater than 0. The velocity is positive. In the graph above it means that the mobile is moving away from O.

-The slope of the tangent line to the curve at P is less than 0. The velocity would be negative. In the graph above, there are no such points, but in this case the particle would be approaching O.

-The slope of the tangent line to the curve is constant at P and all other points. In this case the graph is a straight line and the mobile has uniform rectilinear motion MRU (its speed is constant).

In general, the function v (t) is also a function of time, which in turn can have a derivative. What if it were not possible to find the derivatives of the functions x (t) and v (t)?

In the case of x (t) it could be that the slope - the instantaneous velocity - changes sign abruptly. Or that it would go from zero to a different value immediately.

If so, the graph x (t) would present points or corners at the places of sudden changes. Very different from the case represented in the previous image, in which the curve x (t) is a smooth curve, without points, corners, discontinuities or abrupt changes.

The truth is that for real mobiles, the smooth curves are the ones that best represent the behavior of the object.

The movement in general is quite complex. The mobiles can be stopped for a while, accelerate from rest to have a speed and move away from the starting point, maintain speed for a while, then brake to stop again and so on.

Again they can start again and continue in the same direction. Either operate the reverse and return. This is called varied motion in one dimension.

Here are some examples of calculating the instantaneous velocity will clarify the use of the given definitions:

Solved exercises of instantaneous speed

Exercise 1

A particle moves along a straight line with the following law of motion:

All units are in the International System. Find:

a) The position of the particle at t = 3 seconds.

b) The average speed in the interval between t = 0 s and t = 3 s.

c) The average speed in the interval between t = 0 s and t = 3 s.

d) The instantaneous speed of the particle from the previous question, at t = 1 s.

Answers

a) To find the position of the particle, the law of motion (position function) is evaluated at t = 3:

x (3) = (-4/3).3 ³ + 2. 3 ² ₊ 6.3 - 10 m = -10 m

There is no problem that the position is negative. The sign (-) indicates that the particle is to the left of the origin O.

b) In the calculation of the average velocity, the final and initial positions of the particle are required at the indicated times: x (3) and x (0). The position at t = 3 is x (3) and is known from the previous result. The position at t = 0 seconds is x (0) = -10 m.

Since the final position is the same as the initial position, it is immediately concluded that the mean velocity is 0.

c) The average speed is the ratio between the distance traveled and the time taken. Now, the distance is the module or magnitude of the displacement, therefore:

distance = -x2 - x1- = --10 - (-10) - m = 20 m

Note that the distance traveled is always positive.

v _{m = 20 m / 3 s = 6.7 m / s}

d) Here it is necessary to find the first derivative of the position with respect to time. Then it is evaluated for t = 1 second.

x '(t) = -4 t ² + 4 t + 6

x '(1) = -4.1 ² + 4.1 + 6 m / s = 6 m / s

Exercise 2

Below is the graph of the position of a mobile as a function of time. Find the instantaneous velocity at t = 2 seconds.

Graph of position versus time for a mobile. Source: self made.

Reply

Draw the tangent line to the curve at t = 2 seconds, then find its slope, taking any two points on the line.

To calculate the instantaneous velocity at the indicated point, draw the tangent line to that point and find its slope. Source: self made.

In this example we will take two points that are easily visualized, whose coordinates are (2 s, 10 m) and the cut with the vertical axis (0 s, 7 m):

References

1. Giancoli, D. Physics. Principles with Applications. 6 ^th Edition. Prentice Hall. 22-25.
2. Resnick, R. (1999). Physical. Volume 1. Third edition in Spanish. Mexico. Compañía Editorial Continental SA de CV 21-22.
3. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7 ^ma. Edition. Mexico. Cengage Learning Editors. 23-25.