Conservation of Momentum & Mechanical Energy?

Question

Can some1 explain? Thanks
A 15 gram bullet is fired horizontally into a block of wood with a mass 2.5 kg and embeded in the block. Initally the block of wood hangs vertically and the impact causes the block to swing so that its center rises 15cm . Find the velocity of the bullet just before the impact?

Answer 1

Conservation of momentum and energy are invoked.

Conservation of momentum:

mV = (m + M)v; where m = bullet mass, V = bullet velocity, M = block mass, and v = bullet plus block velocity after embedding in the block. Solve for v, you'll use it in the next equations.

Conservation of energy:

PE = potential energy = (m + M)gh = 1/2 (m + M)v^2 = kinetic (mechanical) energy = KE; where g = 9.81 m/sec^2 the acceleration due to gravity at Earth's surface and h is the height the block and bullet rise above the lowest point of the block just before the impact. v is the same v you found by the conservation of momentum equations.

You can do the math (don't forget to convert the grams to kg and the cm to meters for units consistency).

Key points to remember:

Potential energy equals kinetic energy (PE = KE) when all the kinetic energy is expended while raising something (a mass m) a height (h). Why? Because that KE had to go somewhere (that's what conservation means). And the only energy it could convert into is PE because all motion has ceased at that height (h). (All this presumes 100% efficiency during conversion...which is unrealistic.)

When a fast moving small object embeds in a larger heavier object, the velocity of the two objects together will be less than the fast moving velocity of the small object alone.