How to Find Instantaneous Velocity

Velocity is defined as the speed of an object in a given direction. For common calculations purposes, we use the below formula to find out the velocity.

i.e. V=(S/t)

Where V=Velocity

       S=Distance covered by the object from its origin

       t=Time elapsed

However, this method only gives the object’s average velocity over its path. If we use the calculus method, we can find the object’s velocity at any moment throughout its path. This velocity is known as Instantaneous Velocity. In mathematical term, we can say that the instantaneous velocity is the derivative of the object’s average velocity equation,

i.e. v = (ds)/(dt).

Let’s go step by step to find out how to determine Instantaneous Velocity.

Calculating Instantaneous Velocity

To find out the Instantaneous Velocity, first, we need an equation that tells us its position regarding its movement from the original post at a certain point of time. Which means, we will have S (The displacement) on one side of the equation by itself and t (the time elapsed) on the other hand (*Not mandatorily by itself).

For example: s = 5t2 + 3t + 9 with t = 6

After that take the derivative of the given equation. It is just a different equation which tells you the equation’s slope at any given point in time. To find the derivative of the given equation, we’ll use the most common derivative formula, i.e., If y = a*xn, Derivative = a*n*xn-1. The rule has to be applied to every term on the t-side of the given equation.

Explaining the above rule, you have to go to the t-side of the equation and subtract one from the exponent and multiply the entire term by the original exponent every time you reach a “t”. The terms with no “t” will be increased by 0 and will disappear.

Let’s input this step in our example for better understanding.


s = 5t2 + 3t + 9

=(2)5t(2-1) + (1)3t(1-1) + (0)9t(0-1)

=(2)5t1 + (1)3t0

=10t + 3

Now on the S-side, take the derivative of S concerning “t”. So the S will be replaced by ds/dt.

i.e. ds/dt= 10t + 3

As you can see, you have your new derivative equation. So, it’ll be easy to find the solution. Put the value of the “t”, and you’ll get the instantaneous velocity. Also, remember that Instantaneous velocity is measured by “meter/sec” or “cm/sec”.

As we have given the value of “t” before, which is 6, we’ll just put the value in the equation. Finally,

s=10 x 6 + 3

=60 + 3

=63 m/s

Let’s take another example for better understanding.

The given equation is s = 5t3 – 3t2 + 2t + 9 and t = 4

First, we’ll take our equation’s derivative:

s = 5t3 – 3t2 + 2t + 9

s = (3)5t(3 – 1) – (2)3t(2 – 1) + (1)2t(1 – 1) + (0)9t0 – 1

    =15t(2) – 6t(1) + 2t(0)

    =15t(2) – 6t + 2

Then, put the value of t which is given as 4:

s = 15t(2) – 6t + 2

=15(4)(2) – 6(4) + 2

=15(16) – 6(4) + 2



In this article, we got to know how to find the Instantaneous velocity of an object. Here, we used the derivative rule (If y = a*xn, Derivative = a*n*xn-1) and got the solution successfully. I hope it has cleared all your doubts. Try doing some more problems on your own.

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