wo identical balls A & B each of mass 2 kg & radius R are suspended vertically from inextensible strings as shown. Third ball C of mass 1 kg & radius r = \left(\right. \sqrt{2} - 1 \left.\right) R falls & hits A & B symmetrically with 10 m/s. Speed of both A & B just after the collision is 3 m/s.

wo identical balls A & B each of mass 2 kg & radius R are suspended vertically from inextensible strings as shown. Third ball C of mass 1 kg & radius $r = (\sqrt{2} - 1)$ R falls & hits A & B symmetrically with 10 m/s. Speed of both A & B just after the collision is 3 m/s.

Linked Question 1

Speed of C just after collision is

$(\sqrt{2} - 1) m / s$

2 m/s

5 m/s

$2 \sqrt{2} m / s$

As the balls A & B are constrained to move horizontally, if 'I' be the impulse imparted by ball 'C' to each of A & B, the impulse recieved by ball C from then would be 2I cosθ.
Now, each of ball B & C received impulse 'I' as shown, but moves horizontally as its vertical as its vertical comp. gets balanced by impulse imparted to ball B & C by the respective strings & hence, Icosθ = M_AV_A = M_BV_B

$\Rightarrow I = \frac{M_{A} V_{A}}{cosθ}$ (I = magnitude of Impulse)

Now, for ball C, if its final velocity is V_c' downwards, we have
M_cV_c' = M_cV_c – 2Icosθ

$\Rightarrow V_{c}^{'} = V_{c} - 2 \frac{M_{A}}{M_{C}} V_{A}$

= –2 m/s (–ve sign indicates that $\vec{V_{c}}'$ is directed upwards)

Linked Question 2

The value of coefficient of restitution is

$\frac{1}{2}$

$\frac{1}{\sqrt{2}}$

$(\sqrt{2} - 1)$

$\frac{1}{4}$

$e = \frac{2 \sin 45 ° + 3 \cos 45 °}{10 \sin 45 °} = \frac{1}{2}$

Linked Question 3

Impulse provided by each string during collision is

$6 \sqrt{2} N sec.$

12 N sec.

$3 \sqrt{2} N sec.$

6 N sec.

Impulse provided by each string
Icosθ = 6 N sec.