Conservation of energy + momentum question

[fashafosho]

(From Kaplan) A bloc of mass "m" starts from rest and slides down a frictionless semi-circular track from a height "h". When it reaches the lowest point of the track, it collides with a stationary piece of putty also having mass "m". If the block and the putty stick together and continue to slide, the maximum height that the block-putty system could reach is:
a) h/4
b) h/2
c) h
d) independent of h












The answer is h/4. And the solution goes through a super long substitution (that I don't really get), re-arrangement of kinetic energy and momentum formulas..I was wondering if there is an easier way to do this problem (faster too). Thanks

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[tncekm]

v1=sqrt(2gh)

m1v1 = (m1+m2)v2
v2=v1(m1/(m1+m2))v2 = 1/2 * v1

This means KE has decreased by a factor of 4 since v went down by a factor of 1/2 (KE = 1/2mv^2).

When the KE goes down by a factor of 4 the total E goes down by a factor of 4, and at the max height all KE goes to PE, which means.

h = KE/4mg or, h = (1/4)KE/mg which suggestst that the new height is only 1/4 of the original height.

 

[fashafosho]

Awesome, that simplifies it a lot. Thank you!