A particle of mass m1 makes an elastic head on collision with a stationary particle of mass m2. the fraction of kinectic energy of m1 carried by m2?
1 Answer
Let initial velocity of particle
Since collision is an elastic head on, from Law of Conservation of Momentum we have
#m_1u_1 = m_1 v_1 + m_2 v_2#
#=>m_2 v_2=m_1u_1 - m_1 v_1 # .........(1)
From the Law of Conservation of kinetic energy we have
#1/2 m_1 u_1^2 = 1/2 m_1 v_1^2 + 1/2 m_2 v_2^2# .......(2)
Rewriting (1) as
# m_2v_2=m_1(u_1 - v_1 ) # .......(3)
Rewriting (2) as
# m_2 v_2^2= m_1 (u_1^2-v_1^2)# .......(4)
Dividing (4) with (3)
#v_2=u_1+v_1# .......(2)
Eliminating
#(m_1u_1 - m_1 v_1)/m_2=u_1+v_1#
#=>(m_1u_1 - m_1 v_1)=m_2u_1+m_2v_1#
#=>m_1u_1 -m_2u_1 =m_1 v_1+m_2v_1#
#=>(m_1 -m_2)/(m_1+m_2)u_1 =v_1# .........(6)
Fraction of KE of
#(KE_i-KE_f)/(KE_i)=1-(KE_f)/(KE_i)=1-(1/2m_1v_1^2)/(1/2m_1u_1^2)=1-(v_1/u_1)^2#
Using (6) in above we get
Fraction of KE of
#m_1# carried by#m_2# is
#1-[(m_1 -m_2)/(m_1+m_2)]^2=(4m_1m_2)/((m_1+m_2)^2)#