Calculus > Instantaneous Velocity

Calculus makes finding the **instantaneous velocity** of an object fairly simple. The instantaneous velocity is basically the velocity of an object at a specific point in time. With calculus, you can find it when given only the displacement function, which is a function stating the distance moved of the object. Technically, the instantaneous velocity is the limit of the change in displacement divided by the change in time, as the change in time approaches 0. This is equivalent to the derivative of the displacement function with respect to x, which is also equal to the velocity function.

## Instantaneous Velocity: Example

**Sample question:** Given a displacement function, set the equation up to solve for **velocity**. For the example we will use a simple problem to illustrate the concept.

x(t) = t^{3} + t^{2} + t + 1

v(t) = dx/dt = d/dt (t^{3} + t^{2} + t + 1)

Step 1: Use the **Power Rule** and rule for derivative of constants to solve for the derivative of the displacement function. More complicated functions might necessitate a better knowledge of the rules of differentiation, but these rules work for the example.

v(t) = dx/dt = d/dt (t^{3} + t^{2} + t +1) = 3t^{2} + 2t + 1

Step 2: Now that you have the formula for velocity, you can find the **instantaneous velocity** at any point. For the example, we will find the instantaneous velocity at 0, which is also referred to as the initial velocity.

v(0) = 3*(0^{2}) + 2*(0) + 1 = 1

This indicates the instantaneous velocity at 0 is 1. If you need to find the instantaneous velocity at multiple points, you can simply substitute for t as necessary. For instance, if you needed to find the velocity at 5 as well as 0, just solve for v(5)

v(5) = 3*(5^{2}) + 2(5) + 1 = 3 *25 + 10 + 1 = 86

*Tip:* When performing the operation in typical physics applications, you will need to make sure to include the units of measurement. Velocity is given in units of distance per time.

An interesting note is that at a change in time of 0, there is technically no velocity of the object as motion requires time to pass in order to occur. This is why we take the limit as the change in time approaches 0, instead of simply considering it at 0.

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