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Chapter 3: Problem 80

Rain is falling vertically at a constant speed of \(7.00 \mathrm{~m} / \mathrm{s}\). At what angle from the vertical do the raindrops appear to be falling to the driver of a car traveling on a straight road with a speed of \(60.0 \mathrm{~km} / \mathrm{h} ?\)

Short Answer

Expert verified
Answer: The raindrops appear to be falling at an angle of 67.32° from the vertical to the driver of the car.

Step by step solution

01

Convert the car's speed to m/s

The speed of the car is given in km/h. We need to convert this to m/s in order to compare it directly with the raindrop's speed. To do this, we will use the conversion factor: 1 km = 1000 m and 1 h (hour) = 3600 s (seconds). So, the car's speed is: \(60.0 \mathrm{~km/h} * \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} * \frac{1 \mathrm{~h}}{3600 \mathrm{~s}} = 16.67 \mathrm{~m/s}\).
02

Calculate the relative speed of raindrops

Since the car is moving horizontally, and rain is falling vertically, the relative motion of raindrops will be the result of adding the horizontal and vertical velocity components: \(\text{Relative velocity (Vi)}\) of raindrops = \(\sqrt{(speed\, of\, car)^2 + (speed\, of\, raindrops)^2} = \sqrt{(16.67\, m/s)^2 + (7.00\, m/s)^2} = \sqrt{278.89\, m^2/s^2 + 49.00\, m^2/s^2} = \sqrt{327.89\, m^2/s^2} = 18.11\, m/s\)
03

Find the inclination angle

Now that we have the relative speed of the raindrops, let's find the angle at which they are falling relative to the car's motion. We can use the tangent function along with the horizontal and vertical speed components to solve for the angle (let's call it \(\theta\)): \(\tan(\theta) = \frac{speed\, of\, car}{speed\, of\, raindrops} = \frac{16.67\, m/s}{7.00\, m/s}\), Now, to find the angle \(\theta\), we take the inverse tangent (also called arctangent) of both sides: \(\theta = \arctan(\frac{16.67\, m/s}{7.00\, m/s}) = \arctan(2.38) \approx 67.32^{\circ}\). So, the raindrops appear to be falling at an angle of \(67.32^{\circ}\) from the vertical to the driver of the car.

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Most popular questions from this chapter

A particle's motion is described by the following two parametric equations: $$ \begin{array}{l} x(t)=5 \cos (2 \pi t) \\ y(t)=5 \sin (2 \pi t) \end{array} $$ where the displacements are in meters and \(t\) is the time, in seconds. a) Draw a graph of the particle's trajectory (that is, a graph of \(y\) versus \(x\) ). b) Determine the equations that describe the \(x\) - and \(y\) -components of the velocity, \(v_{x}\) and \(v_{y}\), as functions of time. c) Draw a graph of the particle's speed as a function of time.

A blimp is ascending at the rate of \(7.50 \mathrm{~m} / \mathrm{s}\) at a height of \(80.0 \mathrm{~m}\) above the ground when a package is thrown from its cockpit horizontally with a speed of \(4.70 \mathrm{~m} / \mathrm{s}\). a) How long does it take for the package to reach the ground? b) With what velocity (magnitude and direction) does it hit the ground?

A conveyor belt is used to move sand from one place to another in a factory. The conveyor is tilted at an angle of \(14.0^{\circ}\) from the horizontal and the sand is moved without slipping at the rate of \(7.00 \mathrm{~m} / \mathrm{s}\). The sand is collected in a big drum \(3.00 \mathrm{~m}\) below the end of the conveyor belt. Determine the horizontal distance between the end of the conveyor belt and the middle of the collecting drum.

You want to cross a straight section of a river that has a uniform current of \(5.33 \mathrm{~m} / \mathrm{s}\) and is \(127, \mathrm{~m}\) wide. Your motorboat has an engine that can generate a speed of \(17.5 \mathrm{~m} / \mathrm{s}\) for your boat. Assume that you reach top speed right away (that is, neglect the time it takes to accelerate the boat to top speed). a) If you want to go directly across the river with a \(90^{\circ}\) angle relative to the riverbank, at what angle relative to the riverbank should you point your boat? b) How long will it take to cross the river in this way? c) In which direction should you aim your boat to achieve minimum crossing time? d) What is the minimum time to cross the river? e) What is the minimum speed of your boat that will still enable you to cross the river with a \(90^{\circ}\) angle relative to the riverbank?

The air speed indicator of a plane that took off from Detroit reads \(350 . \mathrm{km} / \mathrm{h}\) and the compass indicates that it is heading due east to Boston. A steady wind is blowing due north at \(40.0 \mathrm{~km} / \mathrm{h}\). Calculate the velocity of the plane with reference to the ground. If the pilot wishes to fly directly to Boston (due east) what must the compass read?
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