Please don\'t not just put the answer I need it explained so I will know how to

Question

Please don't not just put the answer I need it explained so I will know how to do it. You want to design a ball return for a bowling alley. The balls with a mass 5.0g are to come to rest at the top of a ramp that is 20m long ( of which 0.75m is vertical) and has negligible friction. To do this, you use a spring with a spring constant of 15000N/m to start the ball, initially at rest.
Find the distance the spring needs to be pulled each time to launch the ball up the ramp. Find the speed of the ball as it comes off the spring. Find the total work done on the ball through the process.

Explanation / Answer

first we use energy conservation from the potential of the spring to the potential due to gravity

PEs=PEk

(1/2)kx2=mgh

x2=2mgh/k

now plug in what we know (i'm giong to assume the mass was supposed to be 5.0 kg, not 5.0g)

x2=2(5)(9.8)(.75)/15000

x=.07m

Now we set the potential energy of the spring equal to the kinetic energy of the ball

PEs=KE

(1/2)kx2=(1/2)mv2

v2=kx2/m

now plug in what we know

v2=15000(.07)2/5

v=3.834 m/s

Now to find the work done on the ball we say that work is equal to the change in kinetic enery, since the ball is initially at rest the change in kinetic energy will just be the final kinetic energy

W=KE

W=(1/2)mv2

W=(1/2)(5)(3.834)2

W=36.75 joules

Hope that helps

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