Physics help! Got an answer, don’t know why it’s wrong?

Question

The sledder shown in the figure starts at the top of a frictionless hill and slides down into a valley. What initial speed, (v_i), does the sledder need in order to just make it over the next hill? The height of the first hill is 30 m, and the height of the second hill is 42 m.
Here’s what I did so far: 
First, I calculated the velocity the sledder would have at the bottom of the first hill by equating kinetic energy to the potential energy lost. I got 24.25 m/s as the velocity at the bottom.
Next, I determined the energy required for the sledder to go from the valley to the top of the second hill by following the same energy approach. This gave me 28.69 m/s as the velocity needed to reach the top of the second hill.
Then, I set up the equation (24.25 + v = 28.69) to account for the additional speed (v) needed at the start. Solving for (v), I got 4.44 m/s as the initial speed. 
However, this answer is incorrect, and I can’t figure out why. Can someone help me understand where I went wrong?

Anonymous · Accepted Answer

You don't really want the velocity needed at the bottom.  It's energy and not velocity that's conserved.  In this case, you don't even need to know the kinetic energy needed at the bottom.  All you need is kinetic energy at the start that is equal to the potential energy difference between the two hilltops:
(1/2)m(v0)² = mg(42 m - 30 m)
v0 = sqrt(2g12 m)
which I get as about 15.3 m/s.  In the absence of friction, the mg(30 m) work done by gravity on the way down and the -mg(30 m) done on the first 30 m of the way back up cancel out.
The reason you got a bad velocity is that velocity doesn't add in this case.  Adding 15.3 m/s velocity at the top doesn't add 15.3 m/s at the bottom.  Since the sled is already moving, it's average velocity over the 30 m is greater and therefore the time that it is accelerated by gravity is smaller than the starting from rest situation.  You worked it out...that 15.3 m/s difference only adds about 4.4 m/s to the speed at the bottom.

Mozell Dicki · Answer

It seems to me all you need to do is calculate the velocity attained in a drop equal to (42 - 30) = 12m. If it's all frictionless, shouldn't that be the velocity you need?

Anonymous · Answer

KE + PE = Total Energy
The mass will cancel so we can work with:
PE = GH
KE = 0.5 V^2
PE top of first hill = 9.8 * 30 = 294
PE top of second hill = 9.8 * 42 = 411.6
===>
411.6 - 294 = 117.6 worth of KE
KE = 0.5 V^2
117.6 = 0.5BV^2
V^2 = 235.2
V = 15.34 m/s

Anonymous

Physics help! Got an answer, don’t know why it’s wrong?

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