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Velocity in 2D

A person trav...

Question

A person traveling on a straight line moves with a uniform velocity $v_{1}$ for a distance $x$ and with uniform velocity $v_{2}$ for the next equal time. The average velocity $v$ is given by

A

v = \frac{v_{1} + v_{2}}{2}

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B

v = \sqrt{v_{1} v_{2}}

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C

v = \frac{v_{1} v_{2}}{v_{1} + v_{2}}

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D

v = \frac{v_{1} + v_{2}}{v_{1} v_{2}}

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Solution

The correct option is A $v = \frac{v_{1} + v_{2}}{2}$
As shown, the person travels from A to B through a distance S, where first part $S_{1}$ is travelled in time t/2 and text $S_{2}$ also in time t/2.
So, according to the condition : $v_{1} = \frac{S_{1}}{t / 2}$ and $v_{2} = \frac{S_{2}}{t / 2}$
Average velocity = $\frac{T o t a l d i s p l a c e m e n t}{T o t a l t i m e t a k e n} = \frac{S_{1} + S_{2}}{t} = \frac{\frac{v_{1} t}{2} + \frac{v_{2} t}{2}}{t} = \frac{v_{1} + v_{2}}{2}$

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