A particle traversed along a straight line for first half time with velocity $V_{0}$ . For the remaining part, half of the distance is traversed with velocity $V_{1}$ and other half distance with velocity $V_{2}$ . Find the mean velocity of the particle for the total journey.

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Answer 1

Lets total time $= t$
$s_{1} = V_{0} x (\frac{t}{2}) (A t o B)$
$t_{2} = (\frac{3 - S}{2}) / V_{1}$
$t_{2} = (\frac{S - V _{0} ( t / 2 )}{2}) / V_{1}$
$t + 2 = \frac{S}{2 V _{1}} - \frac{V _{0}}{V _{1}} \times \frac{t}{4}$
$(\frac{t}{2} - t_{2}) = \frac{( \frac{S - S _{1}}{2} )}{V _{2}}$
$\frac{t}{2} - (\frac{3}{2 V _{1}} - \frac{V _{0} t}{4 V _{1}}) = \frac{S - S _{1}}{2 V _{2}}$
$\frac{t}{2} - \frac{S}{2 V _{1}} + \frac{V _{0} t}{4 V _{1}} = \frac{3 - V _{0} t / 2}{2 V _{2}}$
$\frac{t}{2} - \frac{3}{2 V _{1}} + \frac{V _{0} t}{4 V _{1}} = \frac{S}{2 V _{2}} - \frac{V _{0}}{4 V _{2}} t$
$t (\frac{1}{2} + \frac{V _{0}}{4 V _{1}} + \frac{V _{0}}{4 V _{2}}) = S (\frac{1}{2 V _{2}} + \frac{1}{2 V _{1}})$
$V_{a r g} = \frac{S}{t} = \frac{2 V _{1} V _{2} + V _{0} ( V _{1} + V _{2} )}{4 V _{1} V _{2}} / \frac{1 4 + V _{2}}{2 V _{1} V _{2}}$
$\frac{S}{t} = \frac{\frac{2 V _{1} V _{2} + V _{0} ( V _{1} + V _{2} )}{2 V _{1} V _{2}}}{\frac{( V _{1} + V _{2} )}{2 V _{1} V _{2}}}$
$V_{a v g} = \frac{V _{0} ( V _{1} + V _{2} ) + 2 V _{1} V _{2}}{2 ( V _{1} + V _{2} )}$

Answer 2

Lets total time

= t

s_{1} = V_{0} x (\frac{t}{2}) (A t o B)

t_{2} = (\frac{3 - S}{2}) / V_{1}

t_{2} = (\frac{S - V _{0} ( t / 2 )}{2}) / V_{1}

t + 2 = \frac{S}{2 V _{1}} - \frac{V _{0}}{V _{1}} \times \frac{t}{4}

(\frac{t}{2} - t_{2}) = \frac{( \frac{S - S _{1}}{2} )}{V _{2}}

\frac{t}{2} - (\frac{3}{2 V _{1}} - \frac{V _{0} t}{4 V _{1}}) = \frac{S - S _{1}}{2 V _{2}}

\frac{t}{2} - \frac{S}{2 V _{1}} + \frac{V _{0} t}{4 V _{1}} = \frac{3 - V _{0} t / 2}{2 V _{2}}

\frac{t}{2} - \frac{3}{2 V _{1}} + \frac{V _{0} t}{4 V _{1}} = \frac{S}{2 V _{2}} - \frac{V _{0}}{4 V _{2}} t

t (\frac{1}{2} + \frac{V _{0}}{4 V _{1}} + \frac{V _{0}}{4 V _{2}}) = S (\frac{1}{2 V _{2}} + \frac{1}{2 V _{1}})

V_{a r g} = \frac{S}{t} = \frac{2 V _{1} V _{2} + V _{0} ( V _{1} + V _{2} )}{4 V _{1} V _{2}} / \frac{1 4 + V _{2}}{2 V _{1} V _{2}}

\frac{S}{t} = \frac{\frac{2 V _{1} V _{2} + V _{0} ( V _{1} + V _{2} )}{2 V _{1} V _{2}}}{\frac{( V _{1} + V _{2} )}{2 V _{1} V _{2}}}

V_{a v g} = \frac{V _{0} ( V _{1} + V _{2} ) + 2 V _{1} V _{2}}{2 ( V _{1} + V _{2} )}