As shown in the figure below, a bullet is fired at and passes through a plece of

Question

As shown in the figure below, a bullet is fired at and passes through a plece of target paper suspended by a massless string. The bullet has a mass m, a speed v before the collision with the target, and a speed (0.476)v after passing through the target. The collision is inelastic and during the collision, the amount of energy lost is equal to a fraction [(0.463)KE_b BC] the kinetic energy of the bullet before the collision. Determine the mass M of the target and the speed V of the target the instant after the collision in terms of the mass m of the bullet and speed v of the bullet before the collision. (Express your answers to at least 3 decimals.)

Explanation / Answer

Let the system consist of the bullet and the paper. Since momentum in the system is conserved. Let mass of bullet be m and mass of paper be M. Let v be initial speed of bullet.

p_f = p_i
M*V + m*0.476v = mv ========> M = 0.524(mv/V)

For conservation of Energy,
E = transferred energy
KE_f - KE_i = energy transferred
1/2m(0.476v)^2 + 1/2M(V)^2 - 1/2mv^2 = -0.463*1/2mv^2

m(0.476v)^2 + M(V)^2 - mv^2 = -0.463*mv^2

MV^2 = 0.310424*mv^2

  0.524(mv/V)*V^2 = 0.310424*mv^2

0.524*mvV =  0.310424*mv^2

V = 0.592*v

and M = 0.524(mv/V) = 0.524*mv/0.592v = 0.885*m

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