Instantaneous Velocity

If you need to know the formula for instantaneous velocity instantaneously, then you will find the required, highlighted in bold below. Further, you will find formula usage, illustrated through examples. Before we go ahead, let me discuss what do we mean by the velocity of an object.

What is Velocity?

Velocity is a vector quantity which is formally defined as the rate of change of distance 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 while velocity is a vector. Speed is, in fact, the magnitude of velocity. When talking about velocity, we specify it according to some fixed frame of reference and its unit is specified as meters/second. While measuring velocity, there are two ways you can specify it. One is by finding the average velocity, while the other is by finding the instantaneous velocity. The average velocity formula is as follows:

Average Velocity = ΔS / ΔT

where ΔS is the distance covered, while ΔT is the time period of travel.

Definition

Average velocity cannot tell you what the changes in velocity of an object were 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. It's the velocity attained by the object in a very short period of time, compared to the time interval for calculation of average velocity. To put it mathematically,

Instantaneous Velocity = Lim_{ΔT → 0} ΔS / ΔT = dS / dT

It is the velocity of the object calculated in the limit that 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 of motion is calculated by substituting for the time variable in the derivative of displacement.

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

Solved Example

Concepts of physics can never be grasped properly unless you solve problems. Let's solve a problem based on the concept of instantaneous velocity.

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

Solution: We know the equation of motion given by :

S = 4t + 6t²

dS / dT _{(t = 10)}. i.e. to calculate it, you must calculate the derivative of displacement with respect to time and substitute t = 10.

dS/dt = d/dt (4t + 6t²) = 4 + 12t

Therefore,

V_{Instantaneous} at (t =10) = 4 + (12 x 10) = 124 meters/sec.

That bullet is traveling at a phenomenal speed apparently.