Hi. I am working out the problems on Conquering the GRE and here is the question I got stuck:

Question:

A pinball machine launch ramp consisting of a spring of force constant k and a 30 degree ramp of Length L. What is the ball's speed immediately after being launched? The ball has m mass and r radius.

Attempt:

Well, to begin with, this is a 3 question problem and this is the second question. From the first question, I could obtain how much the spring needs to be compressed just to make the top of the ramp without rolling back (and friction is sufficient that the ball begins rolling without slipping after lunch).

so compressed distance x is sqrt(m*g*L/k) which I confirm to be correct.

We wanna know the speed of the ball immediately after being launched. So set up the conservation of energy equation such that

0.5*m*v^2 = 0.5*k*x^2 since all the potential energy is converted into kinetic energy.

Simplify it to m*v^2 = k*x^2. We know x. so plug x in.

k*m*g*L/k = m*g*L which is equal to m*v^2.

Hence, v = sqrt(g*L).

And the book says it is wrong with other argument explaining with accounting both transnational rotational energies... I understand their calculation but what I don't understand is where my argument went wrong about setting all potential energy being converted into kinetic.

Can you please help me out?