What is the initial speed?

Question

The "hang time" of a punt is measured to be 4.50 s. If the ball was kicked at an angle of 53.0° above the horizontal and was caught at the same level from which it was kicked, what was its initial speed? (Neglect air resistance.)

Answer 1

The hang time would be determined by the initial vertical velocity component of the ball.  Half of that time would be used in going up, and half coming back down.

The vertical velocity of the ball is slowed down by gravity at the rate of 32ft/s every second, so its vertical speed is slowed down to zero (at the top of its  arc) after 4.50/2 sec = 2.25 seconds.

thus the vertical component of the speed can be found as 32ft/s2x2.25sec=72 ft/s.

Now this is only the vertical component of the velocity. That's where the angle of 53 degrees comes into play.

The actual total velocity would be greater by a factor of sin(53):

total initial velocity = 72 ft/s / (sin(53)) =90.1 ft/s

Hope this helps.