A ball rolls down a ramp that's set on a table, and from there falls off the table. The height of the ramp, length of the ramp, and height from the table are known, and the task is to find the horizontal displacement.

Question

One can use mgh = 1/2mv^2 + 1/2kmv^2 to find the velocity at the bottom of the ramp, but from there, how does the downward angle affect the calculations for horizontal displacement? When assuming the speed is solely horizontal and breaking it into components (x = vt and y = 1/2gt^2, finding t, then using that to find x), the predicted x displacement is far too large, and when trying to factor in the angle the ball was rolling from, the displacements are still too far from the experimental value. How would one find the horizontal displacement correctly, when taking into account how the ball is being "launched" from the table with a known velocity that is not solely horizontal? Any ideas would be helpful! Attached is a picture to illustrate the situation.

Answer 1

You are way over thinking this. It leaves the table at some height h, at a horizontal velocity v.
The question is how long does it take to fall to the floor?
hf=hi+vi*t-1/2 g t^2 but vi in the horizontal direction is zero, so you can solve for time t to tall the distance h.

Now, knowing the time in the air, you calcualte
distance=horizontalvelocity*timeinair where horizontal velocity is the velociyt it left the table with originally: that does not change.