Irodov Problem 1.209

Problem Statement

A gun fires a bullet of mass m with muzzle velocity v. The gun is mounted on a cart of mass M that can slide on a frictionless surface. Find the velocity of the cart after firing.

Given

Gun+cart mass M, bullet mass m, muzzle velocity v (relative to gun). Initially at rest.

Concepts & Formulas

Momentum conservation. Let V_cart = velocity of cart (backward), v_b = bullet velocity in lab. Muzzle velocity: v_b − V_cart = v… wait: muzzle velocity is relative to gun: v_b = v + V_gun = v − |V_cart|. Conservation: 0 = mV_b + MV_cart (taking forward as positive). muzzle: v_b − V_cart = v → v_b = v + V_cart.

Step-by-Step Solution

Step 1: System initially at rest: p_total = 0.
Step 2: After firing: m·v_b + M·V_cart = 0.
Step 3: Muzzle velocity relative to cart: v_b − V_cart = v → v_b = v + V_cart (note V_cart < 0).
Step 4: m(v + V_cart) + M·V_cart = 0 → V_cart = −mv/(M+m). v_b = v − mv/(M+m) = Mv/(M+m).

Worked Calculation

V_cart = −mv/(M+m). V_bullet = Mv/(M+m).

Boxed Answer

V_cart = −mv/(M+m); V_bullet(lab) = Mv/(M+m)

Physical Interpretation

The cart recoils and the bullet goes slower in the lab than muzzle velocity — momentum conservation on a free platform reduces both speeds relative to ground.