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?

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