Show that the average velocity of a particle over an interval over an interval of time is is either is either smaller than and the average speed of the particle over same interval
Huh? <<is either smaller than and the average >>
I think you want to show that the average velocity is smaller or the same as the average speed.
Remember, velocity is a vector, with direction. Average velociyt is the displacement (Final-initial)/time . Average speed does not include intermediate changes in direction.
For instance, consider going from A to B then Back to A.
average veloctiy would be zero (no change in position). Average speed would not be figured that way
To show that the average velocity of a particle over an interval of time is either smaller than or equal to the average speed of the particle over the same interval, we need to understand the definitions of velocity and speed.
Velocity is a vector quantity that describes both the magnitude (speed) and direction of motion. It is calculated as the displacement divided by the time interval taken.
Speed, on the other hand, is a scalar quantity that describes only the magnitude of motion. It is calculated as the total distance traveled divided by the time interval taken.
To prove that the average velocity is smaller than or equal to the average speed, let's consider a simple scenario:
Imagine a particle moving in a straight line. Let's say it moves 10 meters to the east, then turns around and moves 10 meters back to the west, all within a total time interval of 10 seconds.
The average velocity is calculated by finding the displacement of the particle divided by the time taken. In this case, the displacement is zero because the particle ends up at the same position it started. Therefore, the average velocity is 0 meters per second.
The average speed is calculated by finding the total distance traveled divided by the time taken. In this case, the total distance traveled is 20 meters (10 meters east + 10 meters west). Therefore, the average speed is 2 meters per second.
From this example, we can observe that the average velocity (0 m/s) is indeed smaller than the average speed (2 m/s). However, it is also possible for the average velocity to be equal to the average speed in cases where there is no change in direction.
Hence, we have shown that the average velocity of a particle over an interval of time is either smaller than or equal to the average speed of the particle over the same interval.