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.

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.

what is instantaneous velocity

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,

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

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

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

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.

instantaneous velocity from linear position time graph

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:

locating object position

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

tangent line

c] Find the slope of the tangent as the instantaneous velocity is equal to the slope of the tangent:

find the slope of the tangent

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

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

Instantaneous velocity vs Average velocity:

Sr. No.Instantaneous velocityAverage velocity
1It is the velocity of the object at a particular instant of time.It is the ratio of change in position (Δx) of the object and the respective time duration (Δt).
2It is calculated for the smallest interval ‘dt’.It is calculated for the considerable time interval `\Delta `t.
3It describes the motion at a particular instant.It describes the motion within a certain time interval.
4It is calculated as,
`V_{\text{instantaneous}}` = `\frac{dS}{dt}`
The average velocity is calculated as,
`V_{\text{average}}` = `\frac{\Delta S}{\Delta t}`

Solved 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}`

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

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.
instantaneous velocity numerical

Given:
t = 4 seconds

Solution:-

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

locating the point on the curve

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

tangent to the curve

Step 3] Find the slope of the tangent:

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}`

`V_{t=4}` = 0.2 m/s

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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