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a train travels due south at 34 m/s (relative to the ground) in a rain that is blown toward the south by the wind. the path of each raindrop makes an angle of 67° with the vertical, as measured by an observer stationary on the ground. an observer on the train, however, sees the drops fall perfectly vertically. determine the speed of the raindrops relative to the ground.
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Explanation:

Step1: Relate horizontal speed to train

The southward (horizontal) component of the raindrop's speed relative to the ground equals the train's speed, since the train observer sees vertical fall:
$v_{\text{rain, horizontal}} = 34\ \text{m/s}$

Step2: Use trigonometry for total speed

The horizontal speed is the opposite side of the $67^\circ$ angle with the vertical. Use $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$:
$\sin(67^\circ) = \frac{v_{\text{rain, horizontal}}}{v_{\text{rain, ground}}}$
Rearrange to solve for $v_{\text{rain, ground}}$:
$v_{\text{rain, ground}} = \frac{v_{\text{rain, horizontal}}}{\sin(67^\circ)}$

Step3: Calculate the value

Substitute values ($\sin(67^\circ) \approx 0.9205$):
$v_{\text{rain, ground}} = \frac{34}{0.9205} \approx 36.9\ \text{m/s}$

Answer:

36.9 m/s