## Instantaneous Velocity and Speed – Class 11 | Chapter – 3 | Physics Short Notes Series PDF for NEET & JEE

Instantaneous Velocity and Speed: Instantaneous velocity and speed are concepts used in physics and related fields to describe the motion of an object.

Instantaneous velocity refers to the velocity of an object at a specific point in time. It is defined as the rate of change of displacement with respect to time at a particular instant in time. Mathematically, it is the derivative of the position function with respect to time, and can be expressed as:

v = lim Δt→0 (Δx/Δt)

where v is the instantaneous velocity, Δx is the change in position over a very small time interval Δt.

Instantaneous speed, on the other hand, refers to the magnitude of the instantaneous velocity and does not include the direction of motion. It is simply the distance traveled by an object in a given time interval, divided by the duration of that interval.

For example, if a car is traveling at a constant speed of 60 kilometers per hour, its instantaneous speed at any given moment would be 60 kilometers per hour. However, its instantaneous velocity would depend on its direction and could change if the car were to speed up, slow down, or change direction.

## Instantaneous Velocity and Speed

### Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a particular instant in time. It is defined as the rate of change of displacement with respect to time at that specific instant. In other words, it is the limit of the average velocity as the time interval approaches zero.

Mathematically, the instantaneous velocity can be expressed as the derivative of the position function with respect to time:

v = lim Δt→0 (Δx/Δt)

where v is the instantaneous velocity, Δx is the change in position over a very small time interval Δt.

For example, suppose a car is traveling along a straight road and its position is given by the function x(t) = 2t2 + 3t – 1, where x is in meters and t is in seconds. The velocity of the car at a particular instant t is given by the derivative of x with respect to t:

v = dx/dt = 4t + 3

If we want to know the velocity of the car at t = 2 seconds, we can substitute t = 2 into the expression for v to obtain:

v = 4(2) + 3 = 11 m/s

Therefore, the instantaneous velocity of the car at t = 2 seconds is 11 meters per second.

### Instantaneous Speed

Instantaneous speed is the magnitude of the instantaneous velocity of an object at a particular instant in time. It is the rate at which an object is moving without regard to its direction. In other words, it is the distance traveled by an object in a given time interval, divided by the duration of that interval.

The instantaneous speed of an object can be calculated by taking the magnitude of its instantaneous velocity. Mathematically, it can be expressed as:

|v| = lim Δt→0 (|Δx|/Δt)

where |v| is the magnitude of the instantaneous velocity, |Δx| is the absolute value of the change in position over a very small time interval Δt.

For example, suppose a car is traveling along a straight road and its position is given by the function x(t) = 2t2 + 3t – 1, where x is in meters and t is in seconds. The speed of the car at a particular instant t can be calculated by finding the magnitude of its instantaneous velocity:

|v| = |dx/dt| = |4t + 3|

If we want to know the speed of the car at t = 2 seconds, we can substitute t = 2 into the expression for |v| to obtain:

|v| = |4(2) + 3| = 11 m/s

Therefore, the instantaneous speed of the car at t = 2 seconds is 11 meters per second.

## Difference between Instantaneous Velocity and Speed

 Instantaneous speed Instantaneous velocity Instantaneous speed is the magnitude of instantaneous velocity Instantaneous velocity is the change in position taking place at a small change in time Instantaneous speed is a scalar quantity Instantaneous velocity is a vector quantity

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