Find the ratio of velocities of two equal masses in an inelastic collision when one of themasses is stationary (Note : Express your answer in terms of coefficient of restitution e)
Question
Find the ratio of velocities of two equal masses in an inelastic collision when one of themasses is stationary (Note : Express your answer in terms of coefficient of restitution e)
Solution
In an inelastic collision, the coefficient of restitution (e) is defined as the ratio of the final relative velocity to the initial relative velocity.
Let's denote:
- m1 and m2 as the masses of the two objects (which are equal in this case)
- u1 and u2 as the initial velocities of the two objects
- v1 and v2 as the final velocities of the two objects
Given that one of the masses is stationary, we can assume u2 = 0.
The coefficient of restitution is defined as:
e = (v2 - v1) / (u1 - u2)
Substituting u2 = 0, we get:
e = (v2 - v1) / u1
We also know that in an inelastic collision, the total momentum before the collision is equal to the total momentum after the collision. Therefore:
m1u1 + m2u2 = m1v1 + m2v2
Substituting m1 = m2 (since the masses are equal), and u2 = 0 (since one mass is stationary), we get:
u1 = v1 + v2
Substituting this into the equation for e, we get:
e = (v2 - v1) / (v1 + v2)
This is the ratio of the velocities of the two masses in terms of the coefficient of restitution e.
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