Instantaneous velocity formula
Find the Equation of Motion
To be able to compute the velocity of an object at any instant, its equation of motion (the equation establishing the relation of displacement, with time) needs to be figured out.
Velocity is a vector quantity, that's formally defined as the rate of change of position or displacement with time. When stating any vector like the velocity of an object, we talk about direction, as well as magnitude. That's why, speed and velocity are different things. Speed is a scalar (a pure number, specified by magnitude, without direction), while velocity is a vector. To put it simply, speed is the magnitude of velocity. When talking about velocity, we specify it according to some fixed frame of reference and its unit is meters/second. It can be measured in two ways. One is in the form of average velocity, while the other is instantaneous velocity. The formula for the former is as follows.

Average Velocity = ΔS/ΔT

Here, ΔS is the distance covered and ΔT is the time period of travel.

Instantaneous Velocity Formula

Average velocity cannot tell you how the velocity of an object changed at particular instants of time. Instantaneous velocity, as the name itself suggests, is the velocity of a moving object, at a particular instant of time. In mathematical terms, it can be defined in the following way.

Instantaneous Velocity = LimΔT → 0 ΔS/ΔT = dS/dT

It is the velocity of the object, calculated in the shortest instant of time possible (calculated as the time interval ΔT tends to zero). dS/dT is the derivative of displacement vector 'S', with respect to 'T'. The instantaneous velocity at a particular moment is calculated by substituting the corresponding time variable's value, in the first time derivative of the displacement equation.

When driving a car, the speedometer needle or digital display value (showing the speed in Kilometers per hour) on the dashboard, fluctuates at every moment, depending on the speed attained by it. This value, along with the direction of motion, which changes at every instant, is the instantaneous velocity of the car. Theoretically, it should be measured in the shortest time slice possible. That's the reason why the derivative is calculated by assuming ΔT tending to zero.

From the total amount of time taken for the car journey and the distance covered, the average velocity can be calculated, but the instantaneous value will wary over the journey. That is, on an average, your car may have been driven at 50 Km/hr, but at any instant, it may have attained values ranging from 30 Km/hr, 40 Km/hr, or even 60 Km/hr at different instants.

The concept can also be understood in terms of a two-dimensional graph of time (on X-axis), against displacement (on Y-axis). In terms of the graph, instantaneous velocity at a moment, is the slope of the tangent line drawn at a point on the curve, corresponding to that particular instant.

Instantaneous velocity as slope value

Calculus, developed by Sir Isaac Newton and Leibniz, can calculate small changes over time by incorporating the concepts of limit and derivative. The equation above requires you to know how to calculate a derivative. Let me illustrate the use of the formula with some examples.

How to Solve Instantaneous Velocity Problems

Concepts of physics can never be grasped properly unless you solve problems and crank the mathematical machinery underneath it all. Let's solve some problems.

Problem 1: A bullet fired in space is traveling in a straight line and its equation of motion is S(t) = 4t + 6t2. If it travels for 15 seconds before impact, find the instantaneous velocity at the 10th second.

Solution: We know the equation of motion:

S(t) = 4t + 6t2

(S is the displacement or the distance covered.)

dS/dt = d/dt (4t + 6t2) = 4 + 12t

Therefore, VInstantaneous at (t =10) = 4 + (12 x 10) = 124 m/s.

Apparently, that bullet is traveling at a phenomenal speed.

Problem 2: A body is released to fall under the influence of gravity. Its approximate equation of motion is given by S(t) = 4.9 t2. What would be the instantaneous velocity of the body at the fifth second after release?

Solution: In this case, the equation of motion is:

S(t) = 4.8 t2

Instantaneous velocity at t = 5s is given by:

VInstantaneous = [dS/dt]t=5 = [d/dt(4.9 t2)]t=5

= [4.9 x 2 x t]t=5 = 4.9 x 2 x 5 = 49 m/s

Here, the derivative rule used is d/dx(xn) = nxn-1.

Here is a summary of how to go about solving these problems. To find the instantaneous velocity, first the equation of motion (the relation of displacement, with the time and distance variables) needs to be constructed or known. Taking its first derivative, you get the equation for velocity. On substitution of the value of the time variable, you get the required value of velocity at that instant.