Take the general case of an object of mass m_A and velocity v_A elastically stri

Question

Take the general case of an object of mass m_A and velocity v_A elastically striking a stationary (v_B = 0) object of mass m_B head-on. Show that the final velocities v'_A and v'_B are given by v'_A = (m_A - m_B / m_A + m_B) v_B, v'_B = (2m_A / ,_A + m_B)v_A. What happens in the extreme case when m_A is much smaller than m_B? Cite a common example of this. What happens in the extreme case when m_A is much larger than m_B? Cite a common example of this. What happens in the case when m_A = m_B? Cite a common example.

Explanation / Answer

va = initial velocity of mass ma

vb = initial velocity of mass mb

v'a = initial velocity of mass ma

v'b = initial velocity of mass mb

using conservation of momentum

ma va + mb vb = ma va' + mb vb'

ma va = ma va' + mb vb'          eq-1       since vb = 0

for elastic collision

vb' - va'   = va - vb  

vb' - va'   = va

vb' = va' + va             eq-2

using eq-1 and eq-2

ma va = ma va' + mb (va' + va )

(ma - mb) va = (ma + mb) v'a

v'a = (ma - mb) va / (ma + mb)

using eq-2

vb' = ((ma - mb) va / (ma + mb) ) + va  

vb' = 2 ma va / (ma + mb)

b)

ma << mb

v'a = (ma - mb) va / (ma + mb) = - mb va / mb = - va

vb' = 2 ma va /mb

c)

ma >> mb

v'a = (ma - mb) va / (ma + mb) = ma va / ma = va

vb' = 2 ma va /ma = 2 va

d)

v'a = 0

vb' = 2 va

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