Two blocks `m_(1)` and `m_(2)` are pulled on a smooth horizontal surface, and are joined together with a spring of stiffness `k` as shown in Fig. Suddenly, block `m_(2)` receives a horizontal velocity `v_(0)`, then the maximum extension `x_(m)` in the spring is

Options

A. `v_(0)sqrt((m_1m_2)/(m_1 + m_2))`
B. `v_(0)sqrt((2m_1m_2)/((m_1 + m_2)k)`
C. `v_(0)sqrt((m_1m_2)/(2(m_1 + m_2)k)`
D. `v_(0)sqrt((m_1m_2)/((m_1 + m_2)k)`

Answer

D

Explanation

Extension is maximum when velocities become equal.
`implies m_2v_0 = (m_1 + m_2)V implies V = (m_2 v_0)/(m_1 + m_2)` And loss in KE = gain in elastic potential energy.
`implies 1/2 m_2 v_0^2 - 1/2(m_1 + m_2)((m_2v_0)/(m_1 + m_2))^2 = 1/2 kx_m^2 " "implies " "x_(m) = v_(0) sqrt((m_1m_2)/((m_1 + m_2)k))` .