Question

27) The recoil of a shotgun can be significant. Suppose a 3.3-kg shotgun is held tightly by an arm and shoulder with a combined mass of 15.0 kg. When the gun fires a projectile with a mass of 0.038 kg and a speed of 385 m/s, what is the recoil velocity of the shotgun and arm shoulder combination?

Answer 1

Since the initial linear momentum of the system is 0, the linear momentum of the gun-arm system must be equal in magnitude and opposite in direction to the linear momentum of the bullet.

Find the linear momentum of the bullet:

$\begin{gathered} p_{\text{bullet}}=m_{\text{bullet}}v_{\text{bullet}} \\ =(0.038kg)(385\frac{m}{s}) \\ =14.63\operatorname{kg}\cdot\frac{m}{s} \end{gathered}$

If the gun-arm system recoils with a speed v, the total linear momentum of the gun-arm system is:

$(3.3\operatorname{kg}+15.0\operatorname{kg})v$

Since the linear momentum of the gun-arm system must have the same magnitude as the linear momentum of the bullet, then:

$(3.3\operatorname{kg}+15.0\operatorname{kg})v=14.63\operatorname{kg}\cdot\frac{m}{s}$

Isolate v and find its value:

$\begin{gathered} v=\frac{14.63\operatorname{kg}\cdot\frac{m}{s}}{3.3\operatorname{kg}+15.0\operatorname{kg}} \\ =0.79945\ldots\frac{m}{s} \\ \approx0.8\frac{m}{s} \end{gathered}$

Therefore, the recoil speed of the shotgun and arm-shoulder combination is 0.8 meters per second.