Question

A travels at speed 4c/5 toward B, who is at rest. C is between A and B. How fast should C travel so that she sees both A and B approaching him/her at the same speed?​

Answer 1

C should travel at a speed of 2c/5 toward B so that she sees both A and B approaching him/her at the same speed.

Explanation:

→ If there are two particles A and B travelling with a velocity $\boldsymbol{\vec{V}_{A} }$ and $\boldsymbol{\vec{V }_{B}}$ respectively, then the relative velocity of A with respect to B is given as:

       $\boldsymbol{Relative\:Velocity\:of\:'A'\:w.r.t.\:'B'=\vec{V}_{AB}=\vec{V}_{A}-\vec{V}_{B}}$

→ In the given question:

A travels at speed 4c/5 toward B, who is at rest. C is between A and B.

Assume: The direction from A to B is the positive Y-axis and the unit vector along this direction be $\hat{\textbf{\j}}$.

Let the velocity of C be $\boldsymbol{\vec{V}_{C} }$.

∵ B is at rest therefore its velocity $\boldsymbol{(\vec{V }_{B})}$ would be zero.

     $\boldsymbol{\therefore Relative\:Velocity\:of\:'C'\:w.r.t.\:'B'=\vec{V}_{CB}=\vec{V}_{C}-\vec{V}_{B}=\vec{V}_{C}}$

∵ A travels at speed 4c/5 toward B.

∴ The velocity of A = $\boldsymbol{\vec{V}_{A} }$ = (4c/5) $\hat{\textbf{\j}}$

$\boldsymbol{\therefore Relative\:Velocity\:of\:'C'\:w.r.t.\:'A'=\vec{V}_{CA}=\vec{V}_{C}-\vec{V}_{A}=\vec{V}_{C}-(4c/5)\hat{\textbf{\j}}}$

→ Since C sees both A and B approaching him/her at the same speed.

(i) The magnitude of the relative velocity of C with respect to B must be equal to the magnitude of the relative velocity of C with respect to A.

(ii) However the relative velocity of C with respect to B must have the direction opposite to that of the relative velocity of C with respect to A.

$\boldsymbol{\therefore Relative\:Velocity\:of\:'C'\:w.r.t.\:'A'=- Relative\:Velocity\:of\:'C'\:w.r.t.\:'B'}$

                               $\boldsymbol{\therefore (\vec{V}_{C}-(4c/5)\hat{\textbf{\j}})=-(\vec{V}_{C})}$

                                                  $\\\\\boldsymbol{\therefore 2\vec{V}_{C}=(4c/5)\hat{\textbf{\j}}}\\\\\boldsymbol{\therefore \vec{V}_{C}=(2c/5)\hat{\textbf{\j}}}$

∴ The velocity of C ($\boldsymbol{\vec{V}_{C} }$) comes out to be equal to (2c/5) $\hat{\textbf{\j}}$. Here $\boldsymbol{\hat{\textbf{\j}}}$ signifies that the direction of motion of C should be towards B.

Therefore C should travel at a speed of 2c/5 toward B so that she sees both A and B approaching him/her at the same speed.

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