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Question

Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45o to each other. If they travel by different roads, find the rate at which they are being separated.

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Solution

Refer to the figure.
Given OR = distance covered by man A in time t
OS = distance covered by man B in same time t
(Given that both men have started at same time)

Now, As both men are moving with same velocities and time is also same for both men, distance covered will also be same.

Let OA=OB=x

Thus, by definition of velocity, we can write,
v=dxdt

Now, by cosine rule, we can write,

y2=x2+x2+2×(x)×(x)×cos(450)

y2=2x2+2x2×12

y2=2x2+2x2

y2=x2(2+2)

y=x(2+2)

Differentiate both sides w.r.t. t, we get,

dydt=(2+2)dxdt

dydt=(2+2)v

This is rate of separation of two men.

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