Why do we need to look for the speed or velocity at the bottom for this problem?

Question

A rollercoaster cart is pulled to the top of the hill. the cart then moves over the crest at an average speed before it plunges down to its lowest point. From there, it climbs over the second hill. What is the cart's speed as it goes over the top of the second hill?

The equation I used for this one is:

KEtop+PEtop = KEbottom+PEbottom

which is the same as:

1/2mv^2(top)+mgh(top) = 1/2mv^2bottom+mgh(bottom)

cancel m and solve for v(bottom)

v(bottom) = square root of v^2(top)+2g(htop-hbottom)

What scientific law or laws are to be considered for you prove that the velocity needed to be computed as the coaster cart travels a particular speed is the velocity at the bottom?

Answer 1

Unless you are dealing with friction, you need not solve for the speed at the bottom to arrive at the speed at the crest of the second rise, and even then it may be unnecessary:

Let

h1 = elevation of the 1st rise from reference

h2 = elevation of 2nd rise from reference

KE2 + PE2 = KE1 + PE1 - Ef

where Ef is the energy lost to friction.

(1/2)mv2^2 + mgh2 = (1/2)mv1^2 + mgh1 - Ef

v2^2 = v1^2 + 2g(h1 - h2) - Ef

v2 = √(v1^2 + 2g(h1 - h2) - Ef)

If Ef is not given, it can be quite tricky to solve for it, as it is a function of the geometry of the roller coaster.