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Velocity in 2D

A particle mo...

Question

A particle moves half the time of its journey with velocity $u$ . The rest of the half time it moves with two velocities $V_{1}$ and $V_{2}$ such that half the distance it covers with $V_{1}$ and the other half with $V_{2}$ . Find the net average velocity. Assume straight line motion.

\frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}

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\frac{2 u (V_{1} + V_{2})}{2 u + V_{1} + V_{2}}

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\frac{u (V_{1} + V_{2})}{2 V_{1}}

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\frac{2 V_{1} V_{2}}{u + V_{1} + V_{2}}

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Solution

The correct option is A $\frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}$

Average velocity:
$¯ v = \frac{total displacement}{total time}$

$¯ v = \frac{\frac{u T}{2} + 2 d}{T}$

Also, $\frac{T}{2} = \frac{d}{V_{1}} + \frac{d}{V_{2}}$

$\Rightarrow d = \frac{T V_{1} V_{2}}{2 (V_{1} + V_{2})}$

So, $¯ v = \frac{\frac{u T}{2} + 2 \frac{T V_{1} V_{2}}{2 (V_{1} + V_{2})}}{T}$

$¯ v = \frac{u}{2} + \frac{V_{1} V_{2}}{(V_{1} + V_{2})}$

$\Rightarrow ¯ v = \frac{u (V_{1} + V_{2}) + 2 V_{1} V_{2}}{2 (V_{1} + V_{2})}$

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A particle moves half the time of its journey with velocity u. The rest of the half time it moves with two velocities V1 and V2 such that half the distance it covers with V1 and the other half with V2. Find the net average velocity. Assume straight line motion.

The correct option is A u(V1+V2)+2V1V22(V1+V2) Average velocity: ¯v=total displacementtotal time ¯v=uT2+2dT Also, T2=dV1+dV2 ⇒d=TV1V22(V1+V2) So, ¯v=uT2+2TV1V22(V1+V2)T ¯v=u2+V1V2(V1+V2) ⇒¯v=u(V1+V2)+2V1V22(V1+V2)

A particle moves half the time of its journey with velocity $u$ . The rest of the half time it moves with two velocities $V_{1}$ and $V_{2}$ such that half the distance it covers with $V_{1}$ and the other half with $V_{2}$ . Find the net average velocity. Assume straight line motion.