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Two particles , of masses M and 2M , moving as shown , with speeds of 10 m/s and 5 m/s ,

collide elastically at the origin.After the collision , they move along the indicated directions with

speeds $v_{1}$ and $v_{2}$ , respectively. The values of $v_{1}$ and $v_{2}$ are nearly :

Option 1)
6.5 m/s and 6.3 m/s

Option 2)
3.2 m/s and 6.3 m/s

Option 3)
6.5 m/s and 3.2 m/s

Option 4)
3.2 m/s and 12.6 m/s

Answers (1)

best_answer

Elastic Collision in 2 dimension -

$\fn_jvn \vec{P_{i}}= \vec{P_{f}}$

$m_{1}v_{0}\: \: \hat{i}= \left ( m_{1}v_{1} \cos \Theta + m_{2}v_{2} \cos \beta \right )\hat{i}+\left ( m_{1}v_{1} \sin \Theta - m_{2}v_{2} \sin \beta \right )\hat{j}$

- wherein

$u_1$ $u_{1x}=10\times \frac{\sqrt3}{2}\hat{i}$

$u_{1y}=10\times \frac{1}{2}\hat{j}=-5\hat{j}$ $M_1=M$

$u_2$ $u_{2x}= \frac{5}{\sqrt2}\hat{i}$

$u_{2y}= \frac{5}{\sqrt2}\hat{j}$ $M_2=2M$

$v_1$ $v_{1x}=v_1\times \frac{\sqrt3}{2}\hat{i}$

$v_{1y}=v_1\times \frac{1}{2}\hat{j}$

$v_2$ $v_{2x}=v_2\times \frac{1}{\sqrt2}\hat{i}$

$v_{2y}= \frac{v_2}{\sqrt2}(\hat{-j})$

$\Delta P_x=0=>m\times (\frac{10\sqrt3}{2})+(2m\times \frac{5}{\sqrt2})= 2m(\frac{\sqrt3}{2})v_1+m(\frac{v_2}{\sqrt2})$

$=> 5\sqrt3+5\sqrt2=\sqrt3v_1+\frac{v_2}{\sqrt2}$ ................................(1)

$\Delta P_y=0=>-m\times (5)+(2m\times \frac{5}{\sqrt2})= 2m(\frac{v_1}{2})-m(\frac{v_2}{\sqrt2})$

$=>5\sqrt2-5=v_1-\frac{v_2}{\sqrt2}$ ............................................(2)

On adding (1) and (2)

$5(\sqrt3-1)+10\sqrt2=(\sqrt3+1)v_1$

$=> v_1=\frac{5(\sqrt3-1+10\sqrt2)}{\sqrt3+1}\approx 6.5$

$=> v_2\approx 6.3$

Option 1)

6.5 m/s and 6.3 m/s

Option 2)

3.2 m/s and 6.3 m/s

Option 3)

6.5 m/s and 3.2 m/s

Option 4)

3.2 m/s and 12.6 m/s

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