How was the formula obtained?

Question

how was the formula d_{x =} 2√hy obtained? Show your derivation.

Answer 1

When the ball reaches the surface of the table, its speed is

 

v = √(2gh)

 

You get that by equating mgh at the top of the incline with mv²/2 on the surface of the table.  The ball leaves the table with a constant horizontal speed v, and falls a vertical distance y in a time t given by

 

y = gt²/2 ⇒

 

t = √(2y/g)

 

Therefore, it travels a horizontal distance (before hitting the floor) given by

 

d_x = vt = √(2gh) √(2y/g) = 2√(hy)

 

Note both h and y are inside the radical.  End of proof.