Chapter 14, Problem 26A

If a cannonball is fired from tall mountain, gravity changes its speed all along its trajectory. But if it is fired fast enough to go into circular orbit, gravity does not change its speed at all. Why?

To determine

The cause of gravity not changing the speed of the cannonball when it is in circular orbit around earth.

Explanation of Solution

Introduction:

Work done is given as

  $W=Fd\cos \theta$

Where,

  $W=$ Work done

  $\text{F = }$ Force

  $\text{d =}$ Displacement

  $\theta \text{ =}$ Angle between force and displacement

During the motion of the cannonball in circular orbit, the gravitational force acts on the cannonball along the radius of the circular orbit towards the center and perpendicular to the tangent to the circular orbit or the direction of motion of the cannonball. So, the angle between the gravitational force and the direction of motion of the cannonball is always $90°$ .

Hence, the work done on the cannonball by the gravitational force is zero because the value of $\cos 90°$ is zero and hence speed of the cannonball does not change.

Conclusion:

Speed of cannonball does not change because work done by gravitational force on the cannonball is zero.