Instantaneous Speed

The everyday concept of velocity arises when we consider how quick or slow a body moves. Somehow we relate the displacement of the body with the time spent in such displacement. In this section, we will define what is meant in physics by Instantaneous speed and its similarity and difference with instantaneous velocity.

Instantaneous speed

When you travel by car you can look at the speedometer marking the "speed" in every moment of time. Although, normally we just call that value velocity, we have defined previously velocity as a vector. In reality, the car speedometer shows you the magnitude of the instantaneous velocity vector that matches the speed calculated in an instant of time, following a procedure like the one we follow to calculate instantaneous velocity from average velocity, this is, calculating the limit of the average speed when time approaches zero ( $∆ t \to 0$ ).

Instantaneous speed is defined as the limit of the average speed when the considered time interval approaches 0. It is given by the expression:

V = \lim_{∆ t \to 0} \frac{∆ s}{∆ t}

where:

V : Instantaneous speed or simply speed
$∆ s$ : Distance traveled in the interval considered
$∆ t$ : Time taken by the body to complete the motion

The unit of measurement of the instantaneous speed in the International System (S.I.) is meter per second [m/s].

The value of the instantaneous speed coincides with the magnitude of the instantaneous velocity at that point. This is why we use the same letter (V ), but in capital, to denote it. However, do not confuse them: instantaneous velocity or simply velocity is a vector while the instantaneous speed is a scalar. In the following "experiment and learn" you will see that, as we pointed out before, when we take very close points, the value of the average speed approaches the value of the magnitude of the average velocity. At the limit, the values are equal.

Experiment and Learn

Data

Instantaneous speed

The graph shows the trajectory that a body follows, as well as its initial and final position.

Move the time slider to see where the body will be at each time and observe that as the time decreases, the values of the average speed and the magnitude of the average velocity get closer. From this we can deduce that the average speed, when the time approaches zero, is equal to the magnitude of the average velocity, or what is the same, the magnitude of the instantaneous velocity is equal to the instantaneous speed.

Example

Knowing that the distance a body travels as a function of time is given by the following equation:

$S (t) = t^{2} + 5 \cdot t + 1$

Calculate:

a) The average speed during the first 3 seconds.
b) The instantaneous speed of the body.

Solution