In the following apparatus, the moving first cart is allowed to collide with the

Question

In the following apparatus, the moving first cart is allowed to collide with the stationary second cart: The first cart has a mass m_1 and an initial velocity v_o. The second cart has a mass m_2 and is initially at rest. After the collision, the carts stick together (i.e. his is a perfectly inelastic collision). Using the conservation of momentum equation, find an expression for the final velocity of the system assuming the two carts have the same final velocity v_F. Using your answer to Part (a) above, show that the ratio of final kinetic energy to initial kinetic energy is given by the expression: K_F/K_O = m_1/m_1 + m_2

Explanation / Answer

(a) Using the conservation of momentum, an expression for the final velocity of system which is given as :

m1 v1 + m2 v2 = (m1 + m2) v

where, v1 = initial velocity of first cart = v0

v2 = initial velocity of second cart = 0

v = final velocity of two carts = vf

THEN, we get

m1 v0 + m2 (0) = (m1 + m2) vf

vf = m1 v0 / (m1 + m2)

(b) Using an answer of Part-a, the ratio of final kinetic energy to an initial kinetic energy is given by -

K.Ef / K.E0 = (1/2) (m1 + m2) vf2 / (1/2) m1 v02

K.Ef / K.E0 = (m1 + m2) [m1 v0 / (m1 + m2)]2 / m1 v02

K.Ef / K.E0 = [(m1 + m2) m12 v02 / (m1 + m2)2] / m1 v02

K.Ef / K.E0 = m12 v02 / (m1 + m2) m1 v02

K.Ef / K.E0 = m1 / (m1 + m2)

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