In an arrangement for measuring the muzzle velocity of a rifle or pistol, the bullet is fired up at a wooden mass, into which it embeds. The wood is blasted straight up into the air to a measured height h. Assuming negligible losses to friction, write an expression for the velocity in terms of the known masses and height. Use mb for the mass of the bullet, mw for the mass of the wood, h for height and g for gravity.

vw = sqrt ( 2*g*h )

vb = (mb+mw)*sqrt(2*g*h)/mb

Explanation:

Solution:

- First we will use the conservation of momentum just before and after the impact of bullet with wooden mass. We have:

                                   Pf = Pi   ( Pf : final momentum , Pi : initial momentum)

                                   (mb+mw)*vw=mb*vb  

                                   vw=mb*vb/(mb+mw)

- Then we will apply the energy balance on the wood composing of bullet soon after strike position and at the measured height h. We can write the energy balance as:

                                   ΔK.E = ΔP.E

                                   (mb+mw)*g*h=.5*(mb+mw)*vw^2

- Simplify:

                                   vw^2 = 2*g*h

- Plug back in momentum expression and formulate vb:

                                  2*g*h=mb^2*vb^2/(mb+mw)^2

- Simplify:

                                  vb = (mb+mw)*sqrt(2*g*h)/mb

thermal energy is the answer