Question

Two parties A and B are moving with constant velocities $v_1$ and $v_2$.At t = 0,$v_1$ makes an angle ${\theta }_1$ with

the line joining A and B and $v_2$ makes an angle ${\theta }_2$ with the line joining A and B.

(i) Find the condition for A and B to collide.

(ii) Find the time after which A and B will collide.If seperation between them is d at t = 0

• A. Condition = $v_{1}cos{\theta }_1=v_{2}cos{\theta }_2$ time = $\frac{d}{v_{1}sin{\theta }_1+v_{2}sin{\theta }_2}$
• B. Condition = $v_{1}sin{\theta }_1=v_{2}sin{\theta }_2$ time = $\frac{d}{v_{1}cos{\theta }_1+v_{2}cos{\theta }_2}$
• C. Condition = $v_{1}sin{\theta }_1=v_{2}cos{\theta }_2$ time = $\frac{d}{v_{1}cos{\theta }_1+v_{2}cos{\theta }_2}$
• D. Condition = $v_{1}sin{\theta }_1=v_{2}cos{\theta }_2$ time = $\frac{d}{v_{1}sin{\theta }_1+v_{2}sin{\theta }_2}$

Answer 1

The correct option is B

Condition = $v_{1}sin{\theta }_1=v_{2}sin{\theta }_2$

time = $\frac{d}{v_{1}cos{\theta }_1+v_{2}cos{\theta }_2}$

The below is only true if there is no acceleration in picture.

(i) For A and B to collide, their relative velocity must be directed along the line joining them.Therefore

their relative velocity along the perpendicular to this line must be zero.

$\Rightarrow v_A=v_{1}cos{\theta }_1\hat{i}+v_{1}sin{\theta }_1\hat{j}$

$v_B=-v_{2}cos{\theta }_2\hat{i}+v_{2}sin{\theta }_2\hat{j}$

$\Rightarrow v_{BA}=(-v_{2}cos{\theta }_2-v_{1}cos{\theta }_1)\hat{i}+(v_{2}sin{\theta }_2-v_{1}sin{\theta }_1)\hat{j}$

The y-component must be zero.

Thus $v_{1}sin{\theta }_1=v_{2}sin{\theta }_2$

(ii) $v_{App}=v_{1}cos{\theta }_1+v_{2}cos{\theta }_2;t=\frac{d}{v_{app}}=\frac{d}{v_{1}cos{\theta }_2+v_{2}cos{\theta }_2}$