Please help with another physics question!?

Question

A ball is projected horizontally from the top of a building. One second later another ball is projected horizontally from the same point with the same velocity. At what point in the motion will the balls be closest to each other? Will the first ball always be traveling faster than the second ball? How much time passes between the moment the first ball hits the ground and the moment the second one hits the ground? Can the horizontal projection velocity of the second ball be changed so that the balls arrive at the ground at the same time?

Answer 1

The horizontal distance between the balls will always be equal if we ignore air resistance and their horizontal speeds are equal. The instant the first ball is fired, it descends according to
y1(t)=-.5*g*t^2 t greater than 0
the second ball obeys
y2(t)=-.5*g*(t-1)^2 for t greater than or equal to 1

At t=1, y1(t)-y2(t)=-.5*g

for t>1
y1(t)-y2(t)=-.5*g*(2*t+1)

note that this is an increasing function. The first ball gets farther away from the second ball linearly. So they are closest at t=1. What happens after they land cannot be determoned without more information. For example, if they land and stick, say in mud, they are together when the second ball arrives. On the other hand, if an elastic bounce occurs, the first ball will rise up to a point that the two balls have the same separation as when the first ball was launched.

The first ball reaches the ground at
t=sqrt(2*h/g)

Using the quadratic formula and taking the positive root. The quadratic equation is
0=t^2-2*t+(1-2*h/g)

the second ball reaches the ground at
t=1+sqrt(2*h/g), or exactly one second later.

Ignoring air resistance, no change in horizontal velocity will affect the vertical component, so the balls will always strike the ground one second apart.

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Answer 2

THe balls are closest when they land - they should land at the same point. If both must be in motion, then it is the moment of release of the second ball.
The first ball will always be travelling faster (assuming same size, air resistance not a factor) because gravity is a squared function.
One second should pass between when the first ball lands and the second as they are travelling the same path.
No, the horizontal projection velocity can not be changed such that they land at the same time. The horizontal and vertical vectors are independent of each other - gravity will affect them at the same rate so no matter how much velocity is given horizontally the vertical impetus will be the same.

Answer 3

One Second Later