Calculus > How to find Velocity

As **calculus** is the mathematical study of rates of change, and velocity is the measure of the change in position of an object with respect to time, the two come in contact often. Finding the velocity of simple functions can be done without the use of calculus. However, when the graph of a function is curved the velocity is not constant, and performing calculus operations becomes necessary.

You can find the velocity of an object by manipulating **two different functions**. If you are given the formula for the distance moved by the object, you can take the **derivative** and reach the velocity formula. If you are given the formula for the object’s acceleration, you can find the **integral** and again come to the velocity.

## Find velocity given displacement (distance moved by the object)

Step 1: In order to get from the displacement to the velocity, you will take the **derivative** of the displacement with respect to time. First, set up your equation.

x(t) = 4t^{2}+4t+4

dx/dt = d/dt 4t^{2}+4t+4 = v(t)

Step 2: Solve for the derivative. Here you can use the Power Rule, addition rule, and rule for derivative of constants.

d/dt 4t^{2}+4t+4 = 8t+4 = v(t)

*That’s it!*

Tip: The velocity is not constant over time, so *t* makes an appearance. A constant velocity would not have the variable in it, and it would also have an acceleration of 0.

## Find velocity given acceleration.

Step 1: Set up the equation.

a(t) = 10t + 5

v(t) = ∫ a(t) dt = ∫ 10t+5 dt

Step 2: Perform **integration** on a(t). Integration is a somewhat advanced calculus method, so be sure to take a look at articles specifically detailing it if you are unfamiliar with it.

∫ a(t) dt = ∫ 10t+5 dt = 5t^{2}+5t + c = v(t)

That’s how to find velocity!

*Note:* c is a constant of integration that can not be determined without more information. If you are supplied the initial velocity in this example, you can find the constant by setting time equal to 0. Using the above for example:

v(0)= 5(0^{2}) + 5(0) + c = c = v(0)

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