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physics.fisikastudycenter.com - A physics shortcut to collision problem wih conditions the collision should be elastic and the masses of the two bodies involved are the same.


Problem
Given two bodies in an elastic collision above.
What are the velocities of the two bodies after collision?

Answer
The ordinary way used for this question
Step 1
Get the problem data include the signs velocities, in here to the right s (+), and (−) when to the left:
m1 = m2 = 2 kg
e = 1 → elastic collision
v1 = 4 m/s (right)
v2 = − 6 m/s (left)

Step 2
Get the first equation using the conservation of momentum
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
2v₁ + 2v₂ = 2v₁' + 2v₂'
Canceling 2
v₁ + v₂ = v₁' + v₂'
Inserting the data
4 + (−6) = v₁' + v₂'
v₂' + v₁'= −2 ........ (Equation 1)

Step 3
Get the second equation from restitution coefficient e
The coefficient of restitution e for a head-on collision is defined as the ratio of the relative speed after the collision to the relative speed before.
Elastic collision e = 1
where
e = − ^{v₁' − v₂'} / _{v1 − v2}
1 = − ^{v₁' − v₂'} / _{v1 − v2}
v₁ − v₂ = v₂' − v₁'
4 −(−6) = v₂' − v₁'
v₂' − v₁' = `10.......(Equation 2)

Step 4
Combine equations 1 and 2

v₂' + v₁'= −2
v₂' − v₁' = `10
---------------------------------- +
2v₂' = 8
v₂' = 4 m/s

v₂' + v₁'= −2 .
4 + v₁'= −2
v₁'= −2 −4 = −6 m/s

Caution:
Again take a note the conditions for this shortcut:
- elastic collision, do not use in unelastic collision
- m₁ and m₂ are the same in mass, do not use for different masses.