# Instantaneous velocity: Definition, Formula, Examples, Calculation

The velocity is classified as average velocity and instantaneous velocity based on the time interval used to calculate it. In this article, we will go over instantaneous velocity in great detail.

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## What is the Instantaneous velocity?

Instantaneous velocity is the velocity of an object at a specific instant in time. It is an average velocity calculated for the smallest interval of time (dt→0). At a specific instant, it is a ratio of the smallest change in position (d𝑥) to its respective smallest time interval (dt).

During the motion, the object undergoes acceleration and retardation, or sometimes it moves with a constant velocity. Thus only average velocity is not able to describe the motion of the object at every instant of time.

In the above velocity-time graph, the Vaverage indicates the average velocity of the object during its total travel. While Va and Vb indicates the instantaneous velocities of the object at time ta and tb respectively.

## Instantaneous velocity formula:

• By using the method of limits, the instantaneous velocity (IV) of the object at a time ‘t’ is given by,

V =\lim_{\Delta t \to 0} \frac{S_{(t + \Delta t)}-S_{(t)}}{\Delta t}

V(t) =\frac{dS}{dt}

## How to find the Instantaneous velocity?

The following are the two methods to calculate the IV:-

1] Analytical method:

When the equation for the position of the object is provided as a function of time, then in such cases the instantaneous velocity can be calculated by any of the following two methods.

a) Limits: Here the velocity is calculated for the very smallest time interval (Δ t → 0).

V =\lim_{\Delta t \to 0} \frac{S_{(t + \Delta t)}-S_{(t)}}{\Delta t}

b) Differentiation: The instantaneous velocity is also calculated by taking the derivative of the displacement equation with respect to time ‘t’.

V(t) =\frac{dS}{dt}

2] Graphical method:

This method is used to calculate the instantaneous velocity from a position-time graph (Discussed below).

### Instantaneous velocity from a graph:

Following are the two cases to calculate the IV from the position-time graph:-

Case 1] When position-time graph is linear:

As shown in the below figure, if the profile of the position-time graph is linear then it indicates that the object is moving with a constant velocity.

In such cases, the instantaneous velocity at any instant is equal to the average velocity.

Vinstantaneous = Vaverage

Therefore for the above graph, the instantaneous velocity is equal to the slope of the position-time profile.

V_{\text{instantaneous}} = V_{\text{average}} = \text{Slope(1-2)} = \frac{x_{2}-x_{1}}{t_{2}-t_{1}}

Case 2] When position-time graph is non-linear:

The position-time profile in the below figure is non-linear hence the object has different values for instantaneous velocities.

In this case, the instantaneous velocity (IV) is equal to the slope of the tangent drawn to the position-time graph at that instant.

For such a profile, the IV is found by using the following steps:-

a] Locate the position of the object at the time t:

b] At the marked point, draw a tangent line to the curve:

c] Find the slope of the tangent:

\text{Slope(tangent 1-2)} = \frac{x_{2}-x_{1}}{t_{2}-t_{1}}

As the instantaneous velocity is equal to the slope of the tangent, Therefore,

V_{\text{instantaneous}} = \frac{x_{2}-x_{1}}{t_{2}-t_{1}}

## Examples:

1] If the displacement of the object is given by (3t² + 5t) m, then find the velocity of the object at t = 3.5 seconds.

Given:
S = (3t² + 5t) m
t = 3.5 seconds

Solution:-

Method 1: Differentiation

The instantaneous velocity of the object is given by,

V =\frac{dS}{dt}

V =\frac{d}{dt}(3t^{2}+5t)

V = 6t + 5

At t = 3.5 seconds, the instantaneous velocity is given by,

V = 6(3.5) + 5

V = 26 m/s

Method 2: Limit

The instantaneous velocity of the object is given by,

V =\lim_{\Delta t \to 0} \frac{S_{(t + \Delta t)}-S_{(t)}}{\Delta t}

= \lim_{\Delta t \to 0} \frac{[3(t + \Delta t)^{2} +5(t + \Delta t)] – [3t^{2}+5t]}{\Delta t}

= \lim_{\Delta t \to 0} \frac{3t^{2}+6t\Delta t+3\Delta t^{2}+5t+5\Deltat-3t^{2}-5t}{\Delta t}

= \lim_{\Delta t \to 0}\frac{3\Delta t^{2}+6t\Delta t+5\Delta t}{\Delta t}

= \lim_{\Delta t \to 0} 3\Delta t+6t+5

V = 6t + 5

The equation indicates the instantaneous velocity (IV) of the object as a function of time. The IV at t = 3.5 seconds is given by,

V = 6(3.5) + 5

V = 26 m/s

2] The below graph shows the change in position of the object with respect to time t, find the velocity of the object at time t = 4 seconds.

Given:
t = 4 seconds

Solution:-

Step 1] Locate the point on the curve that indicates the position of the object at t = 4 seconds:

Step 2] Draw a tangent to the curve at point a:

Step 3] Find the slope of the tangent:

The slope of the tangent represents the velocity of the object at t = 4 seconds.

V_{t=4} = \text{Slope(1-2)} = \frac{x_{2}-x_{1}}{t_{2}-t_{1}}

From above graph,
x2 = 0.8 m, t2 = 6 seconds, x1 = 0.2 seconds, t1 = 3 seconds

V_{t=4} = \text{Slope(1-2)} = \frac{0.8-0.2}{6-3}

Vt=4 = 0.2 m/s

This is the IV of the object at t = 4 seconds.

## FAQs:

1. What is importance of an Instantaneous velocity?

Instantaneous velocity (IV) is important to describe the motion of the moving object at a particular instant.

2. Why I get instantaneous velocity negative?

The sign of the instantaneous velocity depends on the direction of the motion of the object. Thus if the object is moving in a negative direction, then the I.V. of the object becomes negative.

3. What is an example of instantaneous velocity?

Assuming that the motorcycle is traveling straight, the speedometer on a motorcycle will display the instantaneous velocity of the motorcycle at each instant.

4. How does velocity differ from instantaneous velocity?

Velocity indicates an average value of velocity over a certain distance while instantaneous velocity describes a velocity at a specific instant.

5. Can instantaneous velocity be constant?

The instantaneous velocity changes during acceleration/deceleration and changes in direction, and it remains constant when an object moves with constant speed in the same direction.

6. Can instantaneous velocity become zero?

The instantaneous velocity can be zero. At the instant when motion starts or ends, the object has zero instantaneous velocity.

7. Why is instantaneous velocity not always higher than average velocity?

The average velocity indicates the average value for the velocity of the object. Thus the values for instantaneous velocity are sometimes greater and sometimes lower than the average velocity.

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Pratik is a Graduated Mechanical engineer. He enjoys sharing the engineering knowledge learned by him with people.