Question

A projectile of mass M₁ is fired horizontally from a spring gun that is initially at rest on a friction less surface. The combined mass of the gun and projectile is M₂ If the kinetic energy of the projectile after firing is K, the gun will recoil with a kinetic energy equal to

(A) K

(B) (M₂/M₁)K

(C) (M²₁/M²₂)K

(D) (M₁/(M₂ - M₁))K

(E) √(M₁/(M₂ - M₁))K

Answer 1

Correct option (D) (M₁/(M₂ - M₁))K   

First use the given kinetic energy of mass M1 to determine the projectile speed after. 

K= ½M₁v_1f ² … v_1f = √(2K/M₁) .

Now solve the explosion problem with p_before = 0 = p_after.

Note that the mass of the gun is M₂ – M₁ since M₂ was given as the total mass.

0 = M₁v_1f + (M₂ – M₁)v_2f … now sub in from above for v_1f .

M₁(√(2K/M₁)) = – (M₂ – M₁)v_2f and find v_2f … v_2f = – M₁(√(2K/M₁))/(M₂ – M₁) .

Now sub this into K₂ = ½(M₂ – M₁)v_2f² and simplify