Why does Conservation of Energy and Net Forces produce different answers?

[tkegis]

Homework Statement

Motorcycle in a vertical loop.
Traveling at 50 m/s.
Assume g=10 m/s^2
Find the maximum height of the loop. h=?

Homework Equations

F(net) = F(cent) + F(grav)
F(cent) = mv^2/r
F(grav) = mg

(change) KE = (change) PE
KE = 1/2 mv^2
PE = mgh

The Attempt at a Solution

F(net) solution gives me 500m

F(cent) = F(grav)
mv^2/r = mg
v^2 = gr
v^2/g = r
r = 250m
the height is 500m

Conservation of Energy solution gives me 125m
KE = PE
1/2mv^2 = mgh
(v^2)/(2g) = h
125m = hI found this in an MCAT practice book and they said the correct answer was 500m.
Why does the conservation of energy formula not work here?

 

[Delphi51]

You haven't made the question very clear. Does the motorcycle end up going straight up? Or is it constrained to circular motion on some sort of track? If it is going straight up, your energy solution applies - all of the initial kinetic energy is converted to mgh potential energy. If on the circular track, at the top the motorcycle is moving horizontally so much of its energy is still kinetic and your energy solution does not apply. The centripetal force solution applies if the motorcycle is moving in circular motion at just the right speed to exert zero force on the track at the point of maximum height.

 

[tkegis]

Vertical loop = circular motion

Assuming friction is negligible.
A ball falling down straight or parabolic will reach the ground at the same time.
Does the path taken matter?
Wouldn't the horizontal energy be converted to the vertical to reach the maximum height whether it's launched straight up or along a curve?

 

[Dickfore]

You are not using the Law of conservation of energy correctly. The motorcycle must have a non-zero speed at the highest point in order that the centrifugal force compensates gravity. You assume that the speed of the motorcycle is zero in your "conservation of energy equation".

EDIT:

Also your "net forces" equation involves a speed, but that is not what is given in the problem. Also, think how the height is related to the radius of a vertical loop.

 

[tkegis]

50 m/s is the speed given.
The height would be twice the radius.

A height of 125m seems more realistic than 500m.
Can't argue the physics though. :)

Thank you both.