Question

Three particles each of mass m, can slide on fixed frictionless circular tracks in the same horizontal plane as shown. Particle A moves with velocity $V_0$ and hits particle B elastically. Assuming that B and C are initially at rest and lie along a radial line and the spring is initially relaxed before impact, then

• A. the velocity of B immediately after impact is $V_0$
• B. the velocity of C when the stretch in the spring is maximum is $\frac{2v_0}{5}$
• C. the velocity of B when the stretch in the spring is maximum is $\frac{4v_0}{5}$
• D. the maximum stretch in the spring in the spring is $\sqrt{\frac{m}{5k}}{v}_0$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A the velocity of B immediately after impact is $V_0$
B the velocity of C when the stretch in the spring is maximum is $\frac{2v_0}{5}$
C the velocity of B when the stretch in the spring is maximum is $\frac{4v_0}{5}$
D the maximum stretch in the spring in the spring is $\sqrt{\frac{m}{5k}}{v}_0$

At the time of maximum elongation angular speed of B and C are equal, let speed of B is 2v and C is v, By conserving angular momentum of the system about the centre

$m{v}_{0}2R=m2v\left(2R\right)+mv\left(R\right)$

$v=\frac{2v_0}{5},v_B=\frac{4v_0}{5},v_c=\frac{2v_0}{5}$

Conserving energy of the system $\frac{1}{2}m{v}_0^2=\frac{1}{2}k{x}_{max}^2+\frac{1}{2}m{\left(\frac{4v_0}{5}\right)}^2+\frac{1}{2}m{\left(\frac{2v_0}{5}\right)}^2$

$\therefore x_{max}=\sqrt{\frac{m}{5k}}{v}_0$