Trajectory of A Projectile - Catching Balls

Catching Balls

If a projectile, such as a baseball or cricket ball, travels in a parabolic path, with negligible air resistance, and if a player is positioned so as to catch it as it descends, he sees its angle of elevation increasing continuously throughout its flight. The tangent of the angle of elevation is proportional to the time since the ball was sent into the air, usually by being struck with a bat. Even when the ball is really descending, near the end of its flight, its angle of elevation seen by the player continues to increase. The player therefore sees it as if it were ascending vertically at constant speed. Finding the place from which the ball appears to rise steadily helps the player to position himself correctly to make the catch. If he is too close to the batsman who has hit the ball, it will appear to rise at an accelerating rate. If he is too far from the batsman, it will appear to slow rapidly, and then to descend.

Proof

Suppose the ball starts with a vertical component of velocity of upward, and a horizontal component of velocity of toward the player who wants to catch it. Its altitude above the ground is given by:

where is the time since the ball was hit, and is the acceleration due to gravity.

The total time for the flight, until the ball is back down to the ground, from which it started, is given by:

The horizontal component of the ball's distance from the catcher at time is:

The tangent of the angle of elevation of the ball, as seen by the catcher, is:

The quantity in the brackets will not be zero except when the ball is on the ground, therefore, while the ball is in flight:

The bracket in this last expression is constant for a given flight of the ball. Therefore the tangent of the angle of elevation of the ball, as seen by the player who is properly positioned to catch it, is directly proportional to the time since the ball was hit.