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 Information
- All numerical data are stated in the problem above; symbols are defined as they appear.
Physical Concepts & Formulas
These problems exploit conservation laws. Identify the system, check which quantities (energy, linear momentum, angular momentum) are conserved, and equate initial and final states. Momentum conservation.
- $E_k = \tfrac{1}{2}mv^2$ — kinetic energy
- $\vec{p} = m\vec{v}$ — linear momentum
- $\vec{L} = \vec{r}\times m\vec{v}$ — angular momentum
- $A = \Delta E$ — work–energy theorem
Step-by-Step Solution
Step 1 — Identify the governing principle: We begin by recognising which physical law controls the situation and why it is the correct starting point for this problem.
$$System initially at rest: p_total = 0$$
Step 2 — Set up the relevant equations: Next we write down the equations that follow from that principle, introducing the symbols we will carry through the algebra. After firing: $m \cdot v_b$ + M $\cdot V_cart$ = 0
Step 3 — Apply the given conditions: We now substitute the specific conditions and constraints given in the problem so the equations describe this particular situation. Muzzle velocity relative to cart: $v_b – V_cart$ = v → $v_b$ = $v + V_cart$ (note $V_cart$ < $(0)$
Step 4 — Solve for the required quantity: With the equations specialised, we isolate and solve for the unknown the problem asks us to find.
$$m(v + V_cart) + M \cdot V_cart = 0 → V_cart = -mv/(M+m)$$
$$v_b = v – mv/(M+m) = Mv/(M+m)$$
Worked Calculation
$$V_cart = -mv/(M+m)$$
$$V_bullet = Mv/(M+m)$$
Answer
$$\boxed{V_cart = -mv/(M+m); V_bullet(lab) = Mv/(M+m)}$$
This is the quantity the problem asked for, expressed in terms of the given data: $V_cart = -mv/(M+m); V_bullet(lab) = Mv/(M+m)$.
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. The magnitude of the answer is consistent with everyday physical experience for this class of problem in Irodov’s Part 1 — the result shows how the answer scales with the given quantities. If we doubled the dominant input, the boxed formula tells us exactly how the output would respond, and that scaling is the key physical insight this problem trains. Comparing the answer with the appropriate limiting cases (very small or very large values of the dominant parameter) recovers the familiar Newtonian or intuitive expectation, which is a useful sanity check.
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