First slide

Collisions

Question

Two particles of masses m₁ and m₂ in projectile motion have velocities ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ respectively at time t = 0. They collide at time $t_{0}$ . Their velocities become ${\vec{v}}_{1}'$ and ${\vec{v}}_{2}'$ at time $2 t_{0}$ while still moving in air. The value of $|(m_{1} \vec{v_{1}}' + m_{2} \vec{v_{2}}') - (m_{1} \vec{v_{1}} + m_{2} \vec{v_{2}})|$ is

Moderate

A

Zero
B

$(m_{1} + m_{2}) {gt}_{0}$
C

$2 (m_{1} + m_{2}) {gt}_{0}$
D

$\frac{1}{2} (m_{1} + m_{2}) {gt}_{0}$

Solution

The momentum of the two-particle system, at t = 0 is
${\vec{P}}_{i} = m_{1} {\vec{v}}_{1} + m_{2} {\vec{v}}_{2}$
Collision between the two does not affect the total momentum of the system.
A constant external force $(m_{1} + m_{2}) g$ acts on the system.
The impulse given by this force, in time t = 0 to $t = 2 t_{0}$ is $(m_{1} + m_{2}) g \times 2 t_{0}$
$∴$ |Change in momentum in this interval
$= | m_{1} \vec{v}'_{1} + m_{2} \vec{v}'_{2} - (m_{1} {\vec{v}}_{1} + m_{2} {\vec{v}}_{2}) | = 2 (m_{1} + m_{2}) {gt}_{0}$

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