Simple harmonic motion mass between 2 buildings

Question

I have a question, where I have a mass on top of a building, we give it initial velocity of $\sqrt{gd/2}$ then it starts colliding between 2 buildings, elastic collision. I have to find how many times it will collide between the 2 building before getting to the ground. I know I have to use harmonic motion because everything is conserved since it's elastic collision, and the distance between the buildings won't change. but I don't know how to find the equation I need. some help please?

http://s31.postimg.org/w8pxx9mrv/Untitled.png

this link is the picture and answers I have from the question.

Answer 1

You don't actually have to use harmonic motion,you can solve this question by using equations of motion.

Given that the collision are elastic (coefficient of restitution is 1),i.e,when the ball collides with the building its velocity is reversed.

The point to note here is only the horizontal component is affected by the collision.

Along the vertically downward direction there is gravity acting (which does not affected the horizontal component).
The time taken by the ball to reach the ground is only affected by the vertical downward direction given by the formula $$t=\sqrt{\frac{2h}{g}}$$.

time taken for each collision =$$t_{o}=\frac{d}{V_{0}}$$

hope you can proceed from here

NOTE: while considering the time of collision we have considered horizontal motion only,which is not accelerated

Answer 2

The initial velocity is horizontal. Meanwhile the mass is falling under the influence of gravity. So this is a projectile motion problem and you can use the usual equations for projectile motion. It is not a "harmonic oscillator" problem because the "restoring force" is not proportional to the distance from the equilibrium position. (Where is the equilibrium position??)

Because the mass is "reflected" in the collisions with the buildings you can "unfold" the zig-zag path between the buildings into a continuous parabolic path like a normal projectile (as though the 2nd building were not in the way).

The number of fold creases before the mass hits the ground tells you the number of bounces on the sides of the buildings. Put another way : calculate the range R from the height H of the buildings and horizontal speed $v_0$ which remains constant; then express R as a multiple of d, and round this multiple down to the nearest whole number.