Deriving a formula for KE (rolling + projection)

[foreverlostinclass]

Homework Statement

A ball is rolled down a ramp, then projected a distance R from the end of a curved ramp. The kinetic energy of the ball when it lands is given by the equation: E_k=(gmR^2)/4h, where g is the gravitational acceleration constant (9.81 m/s^2), m is the mass of the ball & h is the height from the ground to the bottom of the ramp.
Edit: Actually it's the kinetic energy when the ball is projected, not when it lands (sorry, misread the description).

Relevant Equations

moment of inertia of a solid sphere: I = 2/5mr^2
K = 1/2mv^2 + 1/2Iw^2
U = mgh

I'm not sure where the equation E_k=(gmR^2)/4h comes from & I also don't really know where to start either :(

 

[kuruman]

According to our rules, to receive help, you need to show some credible effort towards answering the question. How about telling us what you do know and how you would approach this problem?
Please read, understand and follow our homework guidelines, especially item 4, here
https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

Also, it would help if you posted the statement of the problem as was given to you. Note that you are saying that the kinetic energy of the ball is given at the moment of projection but it is not at all clear what you are asked to find.

 

[haruspex]

The kinetic energy of the ball when it lands is given by the equation: E_k=(gmR^2)/4h, where g is the gravitational acceleration constant (9.81 m/s^2), m is the mass of the ball & h is the height from the ground to the bottom of the ramp.
Edit: Actually it's the kinetic energy when the ball is projected, not when it lands (sorry, misread the description).
Relevant Equations: moment of inertia of a solid sphere: I = 2/5mr^2
K = 1/2mv^2 + 1/2Iw^2
U = mgh

It is still not quite true. That formula ignores the rotational KE.
To deduce it, you have to assume the bottom of the ramp is horizontal.
Find

• how long it would take to land, in terms of g and h
• the relationship between that time, R, and the velocity with which it leaves the ramp.