Two particles A and B, move with constant velocities and .  At the initial moment their position vectors are and respectively. The condition for particles A and B for their collision is : Option 1) Option 2) Option 3) Option 4)

P Plabita

As we discussed in

Perfectly Elastic Collision -

Law of conservation of momentum and that of Kinetic Energy hold good.

- wherein

$\frac{1}{2}m_{1}u_{1}^{2}+\frac{1}{2}m_{2}u_{2}^{2}= \frac{1}{2}m_{1}v_{1}^{2}+\frac{1}{2}m_{2}v_{2}^{2}$

$m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}$

$m_{1},m_{2}:masses$

$u_{1},v_{1}:initial \: and\: final \: velocity\: of \: the\: mass\ m_{1}$

$u_{2},v_{2}:initial \: and\: final \: velocity\: of \: the\: mass\ m_{2}$

Let the farticle A and B collide aat time t their Collision, the position vecters both particled should be same at time t.

$r\vec{_{1}}+v\vec{_{1}}t= r\vec{_{2}}+v\vec{_{2}}t$

$r\vec{_{1}}-r\vec{_{2}}= v\vec{_{2}}t-v\vec{_{1}}t=\left | r\vec{_{1}}-r\vec{_{2}} \right |=\left | v\vec{_{2}}-v\vec{_{1}} \right |t$

t= $\frac{\left | r\vec{_{1}}-r\vec{_{2}} \right |}{\left | v\vec{_{2}}-v\vec{_{1}} \right |}$

$\left | r\vec{_{1}}-r\vec{_{2}} \right |=\left | v\vec{_{2}}-v\vec{_{1}} \right |=$ $\frac{\left | r\vec{_{1}}-r\vec{_{2}} \right |}{\left | v\vec{_{2}}-v\vec{_{1}} \right |}$

= $\frac{\vec{r}_{1}-\vec{r}_{2}}{\left | \vec{r}_{1}-\vec{r}_{2} \right |}\:=\:\frac{\vec{v}_{2}-\vec{v}_{2}}{\left | \vec{v}_{2}-\vec{v}_{1} \right |}$

Option 1)

Incorrect Option

Option 2)

Incorrect Option

Option 3)

Incorrect Option

Option 4)

Correct Option

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