Question

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

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

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: \(60.0 \mathrm{~km/h} * \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} * \frac{1 \mathrm{~h}}{3600 \mathrm{~s}} = 16.67 \mathrm{~m/s}\).

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 = \(\sqrt{(speed\, of\, car)^2 + (speed\, of\, raindrops)^2} = \sqrt{(16.67\, m/s)^2 + (7.00\, m/s)^2} = \sqrt{278.89\, m^2/s^2 + 49.00\, m^2/s^2} = \sqrt{327.89\, m^2/s^2} = 18.11\, m/s\)

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.