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Chapter 12: Problem 229

A passenger in an automobile observes that raindrops make an angle of \(30^{\circ}\) with the horizontal as the auto travels forward with a speed of \(60 \mathrm{~km} / \mathrm{h}\). Compute the terminal (constant) velocity \(\mathbf{v}_{r}\) of the rain if it is assumed to fall vertically.

Short Answer

Expert verified
To find the terminal velocity of the rain, two steps calculation is used: first, calculate the velocity of rain as observed by the passenger in the car, and then, convert this velocity to standard units. The final terminal velocity of the rain is approximately \(16.67 \mathrm{~m/ s}\).

Step by step solution

01

Understanding the Situation

First, keep in mind that when rain appears to fall at an angle to a passenger in a moving vehicle, it's due to the forward motion of the vehicle. If the vehicle were stationary, the rain would appear to fall directly downward to the passenger. Thus, the relative velocity of the rain as seen by the passenger is represented by a vector addition of the velocity of the rain and the velocity of the car.
02

Resolving Velocities Vectorially

Let's denote: \( v_c \) as the velocity of the car (60 km/h), \( v_r \) as the velocity of the rain in the stationary frame (which we want to calculate), and \( v_{rp} \) as the relative velocity of the rain with respect to the passenger. Since the velocity of rain observed by the passenger makes an angle of \(30^{\circ}\) with the horizontal, we can use trigonometric function to connect these velocities. From sine ratio, we have: \( \sin(30^{\circ}) = \frac{v_c}{v_{rp}} \)
03

Calculating the Terminal Velocity of Rain

Rearrange the equation from Step 2 to solve for \(v_{rp}\): \( v_{rp} = \frac{v_c}{\sin(30^{\circ})} \). Because the car is stationary with respect to the road and the rain, the velocity of the rain as observed by the car in its frame is assumed to be the actual vertical velocity of the rain, that is \(v_r = v_{rp}\). Thus, substitute the given value of \(v_c = 60 \mathrm{~km} / \mathrm{h}\) into this equation and calculate \(v_r\).
04

Conversion to Appropriate Units

The answer obtained in Step 3 will be in kilometers per hour. Generally, velocities are expressed in meters per second. So, convert the unit from kilometers per hour to meters per second by multiplying the value by the conversion factor \( (1000 \mathrm{~m}) / (3600 \mathrm{~s}) \)

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Most popular questions from this chapter

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