# Instantaneous Velocity: Meaning, Formulas, and Examples

What is the meaning of instantaneous velocity? What is its associated formula? How do you solve problems that are associated with this physics concept? In this article, we answer all these questions for you.

Omkar Phatak

Last Updated: May 22, 2018

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.*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 = Lim**

_{ΔT → 0}ΔS/ΔT = dS/dTIt 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.***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.

*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.

*slope of the tangent line*drawn at a point on the curve, corresponding to that particular instant.

How to Solve Instantaneous Velocity Problems

Problem 1:

A bullet fired in space is traveling in a straight line and its equation of motion is S(t) = 4t + 6t

S(t) = 4t + 6t

(S is the

dS/dt = d/dt (4t + 6t

Therefore, V

Apparently, that bullet is traveling at a phenomenal speed.

^{2}. If it travels for 15 seconds before impact, find the instantaneous velocity at the 10^{th}second.**Solution**: We know the equation of motion:S(t) = 4t + 6t

^{2}(S is the

*displacement*or the*distance covered*.)dS/dt = d/dt (4t + 6t

^{2}) = 4 + 12tTherefore, V

_{Instantaneous}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 t

S(t) = 4.8 t

Instantaneous velocity at t = 5s is given by:

V

= [4.9 x 2 x t]

^{2}. 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 t

^{2}Instantaneous velocity at t = 5s is given by:

V

_{Instantaneous}= [dS/dt]_{t=5}= [d/dt(4.9 t^{2})]_{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(x*.^{n}) = nx^{n-1}*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.