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.

Instantaneous velocity at position vs. time plotFor instance, if we think xt+∆t as xt3 and xt as xt2 and bring t3 closer and closer to t2, 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.

Zero instantaneous velocity at position vs. time plot

Suppose that we have two velocity values v1 and v2. v1=30 m/s and v2=-100 m/s. The velocity v2 is lower than v1 but the speed v2 is higher than v1 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.

Instantaneous Velocity of a Moving Car


Additional Resources