A rubber ball is shot straight up from the ground with speed v_0. Simultaneously

Question

A rubber ball is shot straight up from the ground with speed v_0. Simultaneously, a second ball at hight h directly above the first ball is dropped from rest. a) At what height above the ground do the balls collide? (Use the following as necessary: g, h, v_0) b) What is the maximum value of h for which a collision occurs before the first ball falls back to the ground? (Use the following as necessary: g, and v_0.) c) For what value of h does the collision occur at the instant when the first ball is at its highest point? (Use the following as necessary: g and v_0.)

Explanation / Answer

a) At what height H above the ground do the balls collide?
1st ball: H = v0t - gt2/2 (1)

2nd ball: h - H = gt2/2 (2)

(1) + (2): h = v0t, t = h/v0

H = h - gt2/2 = h - g(h/v0)2/2 = h[1 - gh/(2v02)]

b) What is the maximum value of h for which a collision occurs before the first ball falls back to the ground?

H = 0, so h = 2v02/g

c) For what value of h does the collision occur at the instant when the first ball is at its highest point? (Use the following as necessary: g and v_0.)

at the highest point: t = v0/g, H = v02/g

h = H + gt2/2 = v02/g + g(v0/g)2/2 = 3v02/(2g)

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