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Standard XII

Physics

The 'e' Equation

Two towns T...

Question

Two towns $T_{1}$ and $T_{2}$ are connected by a regular bus service. A scooterist moving from $T_{1}$ to $T_{2}$ with speed of $20 k m h^{- 1}$ notices that a bus goes past it every $21$ minutes in the direction of his motion and every $7$ minutes in the opposite direction. If a bus leaves in either direction every $t$ minutes, the period $t$ is

1.5

minute

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9.5

minute

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6

minute

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10.5

minute

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Solution

The correct option is C $10.5$ minute
Bus (say B1) in the same direction relative speed of bus and cyclist,
$v_{r} = v_{b} - v_{c}$
Distance travelled by B1 relative of cyclist in 21 minutes $= v_{r} \times 21$
Distance covered by bus in $T$ minute $= v_{b} T$
Then, $21 v_{r} = v_{b} T$
Bus in opposite direction (say B2)
Relative speed between B2 and cyclist,
$v_{r} = v_{b} + v_{c}$
Distance travelled by B2 relative of cyclist in 7 minutes $= v_{r} \times 7$
Then $7 v_{r} = v_{b} T$
from, $(i)$ and $(i i)$
$21 v_{r} = 7 v_{r}$ or $3 v_{r} = v_{r}$ or $3 (v_{b} - v_{c}) = v_{b} + v_{c}$ or $v_{b} = 2 v_{c} = 2 \times 20 = 40 k m h^{- 1}$
Thus $T = \frac{7 v_{r}}{v_{b}} = \frac{7 (v_{b} + v_{c})}{v_{b}} = \frac{7 (40 + 20)}{40} = \frac{7 \times 3}{2} = \frac{21}{2} = 10.5 m i n u t e$

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