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12 May, 11:26

The relative kinetic energy loss through a process is defined as ΔΚ Δκ. к, relatine a. Rewrite part (a) in terms of the mass ratio (m, mincoming/mstationary) due to a completely inelastic collision where the second objects is at rest? b. If I want a system that loses 25% of the kinetic energy after a completely inelastic collision, what should the mass ratio be? c. If I want a system that loses 75% of the kinetic energy after a completely inelastic collision, what should the mass ratio be?

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  1. 12 May, 11:36
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    Let mass m₁ is colliding in-elastically with stationary mass m₂ with velocity v₁. Let v₂ be their conjugate velocity after collision.

    Initial KE = 1/2 m₁ v₁²

    Final KE = 1/2 (m₁ + m₂) v₂²

    from conservation of momentum

    v₂ = m₁v₁ / (m₁ + m₂)

    Final KE = 1/2 (m₁ + m₂) m₁²v₁² / (m₁ + m₂) ²

    = 1/2 m₁²v₁² / (m₁ + m₂)

    Loss of KE = ΔK

    = 1/2 m₁ v₁² - 1/2 m₁²v₁² / (m₁ + m₂)

    = 1/2 m₁ v₁² (1 - m₁ / m₁ + m₂)

    = 1/2 m₁ v₁² m₂ / (m₁ + m₂)

    ΔK / K = m₂ / (m₁ + m₂)

    = β / (1 + β)

    where β = m₂ / m₁

    b)

    If ΔK / K =.25

    .25 = β / (1 + β)

    β = 1/3

    c)

    If

    ΔK / K =.75

    .75 = β / (1 + β)

    β = 3
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