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Question
Answer the following question.
Discuss the following as special cases of elastic collisions and obtain their exact or approximate final velocities in terms of their initial velocities.
- Colliding bodies are identical.
- A very heavy object collides on a lighter object, initially at rest.
- A very light object collides on a comparatively much massive object, initially at rest.
Solution
The final velocities after a head-on elastic collision is given as,
`"v"_1 = "u"_1 [("m"_1 - "m"_2)/("m"_1 + "m"_2)] + "u"_2["2m"_2/("m"_1 + "m"_2)]`
`"v"_1 = "u"_1["2m"_1/("m"_1 + "m"_2)] + "u"_2[("m"_2 - "m"_1)/("m"_1 + "m"_2)]`
- Colliding bodies are identical
If m1 = m2, then v1 = u2 and v2 = u1. Thus, objects will exchange their velocities after head on elastic collision. - A very heavy object collides with a lighter object, initially at rest.
Let m1 be the mass of the heavier body and m2 be the mass of the lighter body i.e., m1 >> m2; the lighter particle is at rest i.e., u2 = 0 then,
m1 ± m2 ≅ m1 and `"m"_2/("m"_1 + "m"_2) ~= 0,`
∴ v1 ≅ u1 and v2 ≅ 2u1
i.e., the heavier colliding body is left unaffected and the lighter body which is struck travels with double the speed of the massive striking body. - A very light object collides on a comparatively much massive object, initially at rest.
If m1 is the mass of a light body and m2 is the mass of a heavy body i.e., m1 << m2 and u2 = 0. Thus, m1 can be neglected.
Hence v1 ≅ - u1 and v2 ≅ 0.
i.e., the tiny (lighter) object rebounds with the same speed while the massive object is unaffected.
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