Kinematics

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Question 20 of 308

A person travels xx distance with velocity v1{v}_{1} and then xx distance with velocity v2{v}_{2} in the same direction. The average velocity of the person is vv, then the relation between v,v1v,{v}_{1} and v2{v}_{2} will be

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Answer: (D) 2v=1v1+1v2\dfrac{2}{v}=\dfrac{1}{{v}_{1}}+\dfrac{1}{{v}_{2}}

The time (t1)\left({t}_{1}\right) taken by the person to travel distance xx with velocity v1{v}_{1} is given by
t1=xv1...(1){t}_{1}=\dfrac{x}{{v}_{1}}...\left(1\right)
The time (t2)\left({t}_{2}\right) taken by the person to travel distance xx with velocity v2{v}_{2} is given by
t2=xv2...(2){t}_{2}=\dfrac{x}{{v}_{2}}...\left(2\right)
The formula to calculate the average speed (v)\left(v\right) of the person is given by
v=2xt1+t2...(3)v=\dfrac{2x}{{t}_{1}+{t}_{2}}...\left(3\right)
Substitute the expressions from equations (1) and (2) into equation (3) to obtain the required relation.
v=2x(xv1+xv2)⇒xv1+xv2=2xv⇒2v=1v1+1v2\begin{matrix}v & = & \dfrac{2x}{\left(\dfrac{x}{{v}_{1}}+\dfrac{x}{{v}_{2}}\right)} \\ & \Rightarrow & \dfrac{x}{{v}_{1}}+\dfrac{x}{{v}_{2}}=\dfrac{2x}{v} \\ & \Rightarrow & \dfrac{2}{v}=\dfrac{1}{{v}_{1}}+\dfrac{1}{{v}_{2}}\end{matrix}

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