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Dropping Objects

A man is cycl...

Question

A man is cycling at $4$ $m s^{- 1}$ . To him, rain appears to fall at $30^{o}$ at vertical. If he doubles his velocity, rain appears to fall at $60^{o}$ to vertical. Find the velocity of the rain.

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Solution

$\frac{V_{2}}{4 + V_{1}} = \frac{1}{tan 30 °} V_{2} = (4 + V_{1}) \sqrt{3} ⟶ 1$

When cyclist speed is double then angle changed to $60 °$

$\frac{V_{2}}{8 + V_{1}} = \frac{1}{tan 60 °} V_{2} = \frac{8 + V_{2}}{\sqrt{3}} ⟶ 2$

From $1$ and $2$ equation

$(4 + V_{1}) \sqrt{3} = \frac{8 + V_{2}}{\sqrt{3}} 3 V_{1} + 4 \times 3 = 8 + V_{1} 2 V_{1} = 8 - 12 V_{1} = - 2 m s^{- 1} V_{2} = 2 \sqrt{3} m s^{- 1}$

Velocity of rain $= - 2^i + 2 \sqrt{3} (^j)$

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