Question

Two boys standing at the ends A and B of ground, where AB=a. The boy at B starts running in a direction perpendicular to AB with velocity ${{v}_{1}}$. The boy at A starts running simultaneously with velocity $v$ and catches the other boy in a time t, where t is,
A. $\dfrac{a}{\sqrt{{{v}^{2}}+v_{1}^{2}}}$
B. $\sqrt{\dfrac{{{a}^{2}}}{{{v}^{2}}-v_{1}^{2}}}$
C. $\dfrac{a}{v-{{v}_{1}}}$
D. $\dfrac{a}{v+{{v}_{1}}}$

Answer 1

Hint: Draw a diagram. Assume the time after which the boy at A will be able to catch B. Calculate the distances travelled by both, and complete the right-angled triangle. Use the simple properties of right-angled triangles to find a relation between the known quantities.

Formula Used:
Distance travelled by an object moving with constant velocity in time t is given by,
$s=ut$
Where u is the velocity

Complete step by step answer:
Let us look at the diagram first,

We are going to assume that the boy at A catches the boy at B in time t.

So, the distance travelled by the boy at B is,
$BC={{v}_{1}}t$

The distance travelled by the boy at A is,
$AC=vt$
This forms a right-angled triangle with the sides being,
AB, BC, and CA.

We can simply write the Pythagoras theorem,
$A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}$

Where,
AB, BC, and CA are the sides of the right-angled triangle.

Hence, we can write,
$A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}$
$\Rightarrow {{a}^{2}}+v_{1}^{2}{{t}^{2}}={{v}^{2}}{{t}^{2}}$
$\Rightarrow {{a}^{2}}={{v}^{2}}{{t}^{2}}-v_{1}^{2}{{t}^{2}}$
$\Rightarrow {{t}^{2}}=\dfrac{{{a}^{2}}}{{{v}^{2}}-v_{1}^{2}}$
$\Rightarrow t=\sqrt{\dfrac{{{a}^{2}}}{{{v}^{2}}-v_{1}^{2}}}$

So, t is given by,
$t=\sqrt{\dfrac{{{a}^{2}}}{{{v}^{2}}-v_{1}^{2}}}$

Hence, the correct option is (B).

Note: The boy at A has to run in a direction which should cover both AB and perpendicular distance. That is why we have assumed the boy is running at an angle with the line AB. We can find the angle in the following way,
$\tan \theta =\dfrac{{{v}_{1}}t}{a}$

So, here θ gives the angle with line AB.

You should be able to visualize the problem with the help of the diagram. You can draw a simplistic diagram to avoid any unnecessary calculation mistake.