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Chapter 4: Problem 42

A train travels due south at \(28 \mathrm{~m} / \mathrm{s}\) (relative to the ground) in rain that is blown to the south by the wind. The path of each raindrop makes an angle of \(64^{\circ}\) with the vertical, as measured by an observer stationary on the Earth. An observer on the train, however, sees perfectly vertical tracks of rain on the windowpane. Determine the speed of the drops relative to the Earth.

Short Answer

Expert verified
To find the speed of the raindrops relative to the Earth, substitute the value as calculated in the final step. Calculate \( v_{re} = 28 \mathrm{~m/s} / \cos(64^{\circ}) \) to get the speed of the raindrops relative to the Earth.

Step by step solution

01

Decompose the velocity of the rain

The velocity of the rain can be broken down into two components: vertical and horizontal. The velocity of the rain with respect to Earth, let's call this \( v_{re} \), is the vector sum of these two components. Given that the rain falls at an angle of \( 64^{\circ} \), the horizontal component \( v_{x} \) is \( v_{re}\cos(64^{\circ}) \) and the vertical component \( v_{y} \) is \( v_{re}\sin(64^{\circ}) \).
02

Determine the horizontal velocity

According to the observer on the train, the rain appears to fall vertically, meaning the horizontal component of the rain's velocity is equal to the train's velocity. So, we have \( v_{x} = v_{train} = 28 \mathrm{~m/s} \). Therefore, the rain's speed with respect to Earth is \( v_{re} = v_{x}/\cos(64^{\circ}) \).
03

Calculate the speed of the raindrops

Now substitute the known values into the equation from step 2 to find \( v_{re} \). This will give us the speed of the raindrops relative to Earth.

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Most popular questions from this chapter

A particle leaves the origin at \(t=0\) with an initial velocity \(\overrightarrow{\mathbf{v}}_{0}=(3.6 \mathrm{~m} / \mathrm{s}) \hat{\mathbf{i}}\). It experiences a constant acceleration \(\overrightarrow{\mathbf{a}}=-\left(1.2 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathbf{i}}-\left(1.4 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathbf{j}} .(a)\) At what time does the particle reach its maximum \(x\) coordinate? \((b)\) What is the velocity of the particle at this time? \((c)\) Where is the particle at this time?

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