First slide

The steps in collision

Question

A block C of mass m is moving with velocity v₀ and collides elastically with block A of mass m and connected to another block B of mass 2m through spring of spring constant k. What is the value of k, if x₀ is compression of spring, when velocity of A and B is same? [JIPMER 2017]

Moderate

A

$\frac{m v_{0}^{2}}{x_{0}^{2}}$
B

$\frac{m v_{0}^{2}}{2 x_{0}^{2}}$
C

$\frac{3}{2} \frac{m v_{0}^{2}}{x_{0}^{2}}$
D

$\frac{2}{3} \frac{m v_{0}^{2}}{x_{0}^{2}}$

Solution

Using the law of conservation of linear momentum, we have
$m v_{0} = m v + 2 m v \Rightarrow v = v_{0} / 3$
Using law of conservation of energy, we have
$\frac{1}{2} m v_{0}^{2} = \frac{1}{2} k x_{0}^{2} + \frac{1}{2} (3 m) v^{2}$
where, x₀ is compression in the spring.
$∴$ $m v_{0}^{2} = k x_{0}^{2} + (3 m) \frac{v_{0}^{2}}{9}$
$\Rightarrow$ $k x_{0}^{2} = m v_{0}^{2} - \frac{m v_{0}^{2}}{3} \Rightarrow k x_{0}^{2} = \frac{2 m v_{0}^{2}}{3}$
$∴$ $k = \frac{2 m v_{0}^{2}}{3 x_{0}^{2}}$

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