Notes On Instantaneous Velocity and Speed - CBSE Class 11 Physics
The instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally small. It represents the rate of change of position with respect to time at that instant of time. We can obtain the value of instantaneous velocity either from the x-t graph or numerically using differential calculus if the relation between x and t is known for the motion. Let’s understand how to calculate instantaneous velocity using an x-t graph. We first mark a point P on the graph at which the instantaneous velocity of the body has to be found. We mark two points P1 and P2 on either side of P. Join points P1 and P2. Find Δt and Δx. When Δt is reduced, we get lines Q1Q2, etc. On further decreasing the value of Δt, we finally get one line, which is the tangent to the curve at point P. The slope of this tangent represents the instantaneous velocity at point P. Slope is represented by tan θ  as shown on the graph. This graphical method of finding the instantaneous velocity at any point is cumbersome.  Therefore, we generally use the numerical method, using calculus, for finding the instantaneous velocity at any point.   x = f(t)   v = $\underset{∆\text{t}\to \text{0}}{\text{lim}}\text{}\frac{∆\text{x}}{∆\text{t}}$   v = $\frac{\text{dx}}{\text{dt}}$ When a body moves in a straight line, the magnitude of its velocity is equal to its speed. The speedometer reading of a vehicle gives the instantaneous velocity at that particular moment.

#### Summary

The instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally small. It represents the rate of change of position with respect to time at that instant of time. We can obtain the value of instantaneous velocity either from the x-t graph or numerically using differential calculus if the relation between x and t is known for the motion. Let’s understand how to calculate instantaneous velocity using an x-t graph. We first mark a point P on the graph at which the instantaneous velocity of the body has to be found. We mark two points P1 and P2 on either side of P. Join points P1 and P2. Find Δt and Δx. When Δt is reduced, we get lines Q1Q2, etc. On further decreasing the value of Δt, we finally get one line, which is the tangent to the curve at point P. The slope of this tangent represents the instantaneous velocity at point P. Slope is represented by tan θ  as shown on the graph. This graphical method of finding the instantaneous velocity at any point is cumbersome.  Therefore, we generally use the numerical method, using calculus, for finding the instantaneous velocity at any point.   x = f(t)   v = $\underset{∆\text{t}\to \text{0}}{\text{lim}}\text{}\frac{∆\text{x}}{∆\text{t}}$   v = $\frac{\text{dx}}{\text{dt}}$ When a body moves in a straight line, the magnitude of its velocity is equal to its speed. The speedometer reading of a vehicle gives the instantaneous velocity at that particular moment.

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