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?

Explanation:

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

50 inches should be the resultant vector

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