How to Calculate Instantaneous Velocity

Table of Contents

Due to the continuous motion of the objects through dimensions of space and time, it becomes imperative to calculate the instantaneous velocity. Instantaneous velocity is the mean velocity between two points on a specific path such that the time between the two points approaches zero. It is the quantity that determines the speed of an object. The speed at which an object is moving at any instant anywhere along its path. It is also known as simply velocity, which gives the velocity of an object at any whole point. Like average velocity, the instantaneous velocity is a vector with a direction and magnitude.

Instantaneous refers to the time, while velocity relates to motion. These two are not compatible terms. Perhaps you mean “infinite velocity” or “zero velocity”? Zero and Infinite velocities are problematic because they require a fixed absolute frame of reference in time and space, but we operate out of a relativistic frame. As such, a given object may be in a static position or stationary relative to one observer but in a moving position relative to another. Simultaneity is a relativistic idea that Changes with motion.

This article will depict the instantaneous velocity formula. Moreover, it will show you how instantaneous velocity can be calculated step by step using the base and derived physical quantities and stress on its practical applications.

Instantaneous Velocity Formula

“Instantaneous velocity is total displacement divided by total time as the time interval approaches zero. The displacement also approaches zero but the limit of the ratio of displacement to time is non-zero and is called instantaneous velocity.” It is determined in a very similar way as average velocity, but here the time period is converged


  • Δt is an extremely small time interval.
  • Vi is the instantaneous velocity.
  • S is the displacement covered.
  • T is the time taken for the displacement covered.

What is Instantaneous Velocity

The average velocity over a time interval is computed as the displacement of the object divided by the time interval. Since the velocity is non-uniform in most practical cases, like driving a car, it drives us to explore the rate of change of position at a particular instant, the instantaneous velocity. In another way, Instantaneous velocity is defined as “The velocity of a moving object at instant time of interval. It is also being defined as the rate of change of position for a very small time interval (almost zero).” It is represented using SI unit m/s and is a vector quantity.

If the object possesses uniform velocity, then it becomes the instantaneous velocity of the object. Instantaneous velocity can be shown as a continuous function of time and gives the rate of change of displacement during a particle’s motion at any point in time. For example, the speedometer in your vehicle tells the instantaneous velocity throughout driving.

Importance of Calculating Instantaneous Velocity

The need to calculate the instantaneous velocity of any object is to instantly calculate the exact or accurate value of its velocity. The motions of the objects are so irregular that calculating their velocity for any time interval will always give their average velocities and not the exact velocity value. Therefore, by using instantaneous velocity equations, one can get more accurate or precise values during calculations. Shedding light on what instantaneous velocity is has highlighted its importance in practical situations like collisions. If we need to know the momentum of an object right just before a collision, we shall look at instantaneous velocity.

Calculation Step by Step using the Formula

  1. The change in displacement of the object concerning time is represented in an equation in terms of time.
  2. Take the derivative of the equation of displacement given in terms of time to get the velocity equation.

                          Velocity (V) = ds/dt

  1.  The instantaneous velocity can be found at any instant by substituting the value of time (t)   in the obtained equation (after taking derivative).

Example 1

The motion of any vehicle is described by the equation (6t2 + 2t+4) m. The instantaneous velocity can be calculated any given value of time by the following steps:

How to Calculate Instantaneous Velocity

Tips for Solving Instantaneous Velocity

  • The equation which depicts the relationship between y (displacement) and x (time) might be relatively simple, like, for instance, y= 12x + 5. In this case, the slope is uniform, and it is not required to find a derivative to find the slope, which is 12 (y=mx+c).
  • Displacement is the distance covered but in a set direction. Therefore, it is a vector quantity. Displacement can be negative, but distance will only have a positive magnitude.
  • To find the rate of change of velocity over time, use the method described above to get a derivative for your displacement function. Then, take another derivative of the already obtained derivative equation. This will formulate an equation for finding acceleration at a given time.

Graphical Solution of Instantaneous Velocity

Instantaneous Velocity Formula is used to determine the rate of change of displacement of the given body at any specific instant of time. “The limit of the average velocity as elapsed time reaches zero, or the derivative of displacement x with respect to time t, is the instantaneous velocity of an object.”

Like average velocity, the instantaneous velocity is a derived quantity obtained using dimensions of displacement per unit time. The instantaneous velocity at a specific time t0 is the slope of the displacement function x(t) at t0 illustrated graphically. It is the rate of change of the displacement function. The figure below shows how the average velocity v=Δx/Δt between two times converges to obtain the instantaneous velocity . This velocity is shown at time t0, which occurs at the maximum point of the displacement function on the graph. Moreover, It can also occur at the minimum point on the graph as the slope of the graph is zero at these points.

At times, t1, t2, and so on, the slopes of the graph are either positive or negative, so the instantaneous velocity is not zero. Therefore, the zeros of the velocity function give the maximum and minimum of the displacement function.

The instantaneous velocity has been defined as the slope of the tangent line at a given point in a graph of position versus time. The average velocities v= Δx/Δt = (xf−xi)/(tf−ti) between times Δt=t6−t1, Δt=t5−t2, and Δt=t4−t3 are shown in figure.At t=t0, the average velocity approaches that of the instantaneous velocity.

 Example 2

Consider the motion of a vehicle and its displacement function is given as x(t) = 3t − 3t2 (m). Compute the instantaneous velocity at t = 0.25 s just before the collision, t = 0.50 s during the collision, and t = 1.0 s after the collision?

Take the derivative first as instantaneous velocity is the derivative of the displacement function.

V(t) = dx(t) / dt = 3 – 6t m/s

At t = 0.25 s

V (0.25) = 3 – 6(0.25) = 1.5 m/s

Similarly, substitute the values for time t in the derivative to obtain the instantaneous velocities.

At t = 0.5 s

V (0.5) = 3 – 6(0.5) = 0 m/s

At t = 1 s

V (1) = 3 – 6(1) = -3 m/s

The solution can also be represented in the graphical form shown below.


1Differentiate between instantaneous and average velocity?Instantaneous velocity is at a certain moment, whereas the average is the mean velocity of the entire time span.
2When is instantaneous velocity equal to the average velocity?The average velocity and instantaneous velocity will be the same if the object moves with a uniform velocity. The change in displacement is the same per unit time.
3Define instantaneous speed?The speed of the object at a particular moment or instant of time is called instantaneous speed. It is a scalar quantity.
4State the relationship between instantaneous speed and instantaneous velocity?Instantaneous speed is always the magnitude of the Instantaneous velocity.

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