Problem Statement
Two balls undergo a perfectly inelastic collision. Ball 1 (mass m) moves at $v_0$; ball 2 (mass m) is at rest. Find the fraction of kinetic energy lost.
Given Information
- $v_2 = 0$ — speed
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.
- $E_k = \tfrac{1}{2}mv^2$ — 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.
$$v’ = v_0/2$$
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.
$$KE_i = (1/2)mv_0^{2}$$
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.
$$KE_f = (1/4)mv_0^{2}$$
Step 4 — Solve for the required quantity: With the equations specialised, we isolate and solve for the unknown the problem asks us to find.
$$Fraction lost = (1/2)$$
Worked Calculation
$$\Delta KE/KE_i = 1/2$$
Answer
$$\boxed{Fraction of KE lost = 1/2}$$
This is the quantity the problem asked for, expressed in terms of the given data: $Fraction of KE lost = 1/2$.
Physical Interpretation
Half the initial kinetic energy is converted to heat and deformation in a perfectly inelastic collision between equal masses — the maximum possible loss for this mass ratio. 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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