# Instantaneous velocity: Definition, Formula, Solved Examples

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.

Contents

## What is the instantaneous velocity?

Instantaneous velocity is the velocity of an object at a specific instant of time. At a specific instant, the instantaneous velocity is a ratio of the smallest change in position (d𝑥) to the 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 V_{\text{average}} indicates the average velocity of the object during its total travel. While the V_{a} and V_{b} indicates the instantaneous velocities of the object at time t_{a} and t_{b} respectively.

## Instantaneous velocity equation:

By using the method of limits, the instantaneous velocity of the object at time ‘t’ is given by,

In the form of a differentiation, the instantaneous velocity is given by,

## How to find the instantaneous velocity?

Following are the two methods to find the instantaneous velocity:-

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 (\Delta t \to 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.

## How to find instantaneous velocity from a graph?

Following are the two cases to find the instantaneous velocity 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.

V_{\text{instantaneous}} = V_{\text{average}}

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

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

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

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

For such a profile, the instantaneous velocity 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 as the instantaneous velocity is equal to the slope of the tangent:

V_{\text{instantaneous}} = Slope(tangent 1-2)

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

## Solved examples:

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

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}

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

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

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

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

V = 6t + 5

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

V = 6(3.5) + 5

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 second.

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

From above graph,

x_{2} = 0.8 m, t_{2} = 6 seconds, x_{1} = 0.2 seconds, t_{1} = 3 seconds

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

This is the instantaneous velocity of the object at t = 4 seconds.

## FAQs:

1. Why Instantaneous velocity is important?

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

2. Can instantaneous velocity be 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 instantaneous velocity of the object becomes negative.

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