CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 4, Exercise 2 Solution
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A Perfectly Inelastic Collision: Finding Velocity and Lost Energy

PROBLEM
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A 5 kg cart moving at 4 m/s collides with a stationary 15 kg cart,
and the two carts stick together (a perfectly inelastic collision).
Find their shared final velocity, then calculate how much kinetic
energy was lost in the collision.

SOLUTION
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Step 1: Find the shared final velocity using conservation of
momentum.

    m_A v_A + m_B v_B = (m_A + m_B) v_final

    (5)(4) + (15)(0) = (5 + 15) v_final
    20 + 0 = 20 v_final
    v_final = 20 / 20
    v_final = 1 m/s

Step 2: Find the kinetic energy before the collision.

    KE_before = 1/2 m_A v_A^2 + 1/2 m_B v_B^2
    KE_before = 1/2 (5)(4^2) + 1/2 (15)(0^2)
    KE_before = 1/2 (5)(16) + 0
    KE_before = 40 J

Step 3: Find the kinetic energy after the collision.

    KE_after = 1/2 (m_A + m_B) v_final^2
    KE_after = 1/2 (20)(1^2)
    KE_after = 10 J

Step 4: Find the energy lost.

    Energy lost = KE_before - KE_after
    Energy lost = 40 - 10
    Energy lost = 30 J

ANSWER: The shared final velocity is 1 m/s. 30 J of kinetic energy
was lost in the collision (converted to heat, sound, and
deformation).

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WHY THIS WORKS AS AN ANSWER

Momentum (a vector quantity, mv) is conserved exactly in every
collision, so Step 1 correctly finds the final velocity using only
momentum. Kinetic energy (a scalar quantity, 1/2 mv^2) is NOT
generally conserved in an inelastic collision, which is why Steps 2-4
calculate it separately, before and after, rather than assuming it
stays the same. The 30 J difference (75% of the original kinetic
energy) is real energy that went somewhere else entirely - into the
sound of the impact, heat generated at the point of contact, and any
permanent deformation of the coupling mechanism - exactly the warning
the chapter gives about momentum and kinetic energy being two
genuinely separate conservation questions, not one.