**INSTANTANEOUS VELOCITY FORMULA**

## Instantaneous Velocity Formula

Instantaneous velocity is the velocity at a particular moment in time and it is defined by the slope of the tangent line at that point on the position *vs.** *time graph. The instantaneous velocity formula is

\({ v }_{ t }=lim_{ \Delta t\rightarrow 0 }\frac { { x }_{ { t }+\Delta t }-{ x }_{ t } }{ \Delta t } =\frac { dx }{ dt } \).

In this formula, dx/dt is the first derivative of position versus time. The notation lim _{∆t–>0} means ∆x/∆t ratio is to be determined in the limit that ∆t goes to 0. ∆t cannot be simply set 0 since it would result undefined number for ∆x/∆t. As ∆t approaches 0, the ratio ∆x/∆t approaches some definite value and this is instantaneous velocity at that particular instant.

For instance, if we think x_{t+∆t} as x_{t3 }and x_{t} as x_{t2} and bring t_{3} closer and closer to t_{2}, then the time between them goes to 0 and the angle of the slope line changes as shown in the figure.

Following relationship exist between dx/dt and the slope.

- v= dx/dt >0, the slope is positive
- v=dx/dt = 0, the slope is 0
- v=dx/dt < 0, the slope is negative.

If we look at the same plot again, the times that velocity equals 0 are the points where dx/dt and the slope is 0.

Suppose that we have two velocity values v_{1} and v_{2}. v_{1}=30 m/s and v_{2}=-100 m/s. The velocity v_{2} is lower than v_{1} but the speed v_{2} is higher than v_{1} speed. Instantaneous speed is the magnitude of the instantaneous velocity. It s a scalar quantity and it is always positive.

\({ v }_{ 1 }>{ v }_{ 2 }\quad ,\quad \left| { v }_{ 1 } \right| <{ \left| { v }_{ 2 } \right| } \).

## Instantaneous Velocity Example

In a simpler way, let’s assume you drove a car along a straight road for 100 km to the right in 4 hours as shown in the animation below. During this period, you were accelerating and decelerating so your speed is changing at every moment in time. The average velocity is 100 / 4 = 25 km/h to the right which doesn’t necessarily equal to instantaneous velocities at particular points in the trip.

## Additional Resources

## Reference

- Giancoli, Douglas C. Physics: Principles with Applications (7th Edition) – Standalone book. Pearson, 2016.