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Chapter 2: Problem 192

Rain is falling vertically with a speed of \(20 \mathrm{~ms}^{-1}\) relative to air. A person is running in the rain with a velocity of \(5 \mathrm{~ms}^{-1}\) and a wind is also blowing with a speed of \(15 \mathrm{~ms}^{-1}\) (both towards east). Find the cotangent of the angle with the vertical at which the person should hold his umbrella so that he may not get drenched.

Short Answer

Expert verified
The cotangent of the angle 胃 at which the person should hold his umbrella so that he may not get drenched is 1.

Step by step solution

01

Determine the velocity components

First, let's determine the velocity components of the rain (V_r), person (V_p), and wind (V_w) w.r.t. their axes. The rain is falling vertically, so its velocity component along the y-axis will be -20 m/s (negative for downward direction), and the component along the x-axis will be 0. V_r = (0, -20) The person is running towards the east, so the x-component of his velocity will be 5 m/s, and the y-component will be 0. V_p = (5, 0) Likewise, the wind is also blowing towards the east, so its x-component will be 15 m/s, and the y-component will be 0. V_w = (15, 0)
02

Calculate the cumulative velocity of the rain with respect to the person

Now, let's find the cumulative velocity of the rain w.r.t. the person and wind. To do this, we subtract the combined person's velocity (V_p) and the wind's velocity (V_w) from the rain's velocity (V_r) for both components. V_r_cumulative = V_r - (V_p + V_w) The x-component of the cumulative velocity: 0 - (5 + 15) = -20 m/s The y-component of the cumulative velocity: -20 - (0 + 0) = -20 m/s V_r_cumulative = (-20, -20)
03

Find the angle 胃

We use the components of the velocity vector to find the angle 胃. In the right triangle formed by the cumulative velocity vector and its components, the cotangent of the angle 胃 is the ratio of the adjacent side to the opposite side. \(\cot{\theta} = \frac{Adjacent}{Opposite}\) \(\cot{\theta} = \frac{-20}{-20}\) \(\cot{\theta} = 1\) The cotangent of the angle 胃 at which the person should hold his umbrella so that he may not get drenched is 1.

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