Question

A car is moving at a constant \(19.3 \mathrm{~m} / \mathrm{s}\), and rain is falling at \(8.9 \mathrm{~m} / \mathrm{s}\) straight down. What angle \(\theta\) (in degrees) does the rain make with respect to the horizontal as observed by the driver?

Short answer

Answer: The angle the rain makes with respect to the horizontal as observed by the driver can be found by calculating the arctangent of the ratio of the rain's vertical velocity to the car's horizontal velocity, and then converting the value from radians to degrees: \(\theta (degrees) = \arctan(\frac{8.9}{19.3}) * \frac{180^{\circ}}{\pi}\)

Step-by-step solution

Identify the components of the rain's velocity

As observed by the driver, the rain is falling at an angle because the car is moving. We need to consider both the horizontal and vertical components of the rain's velocity. Horizontal component:\(~v_{h}~\)is equal to the car's velocity, which is \(19.3\mathrm{~m}/\mathrm{s}\). Vertical component:\(~v_{v}~\)is the rain's vertical velocity, which is \(8.9\mathrm{~m}/\mathrm{s}\).

Calculate the resultant velocity as the Pythagorean sum of the horizontal and vertical components

To find the angle, we need the magnitude of the resultant velocity of the rain, which can be calculated using the Pythagorean theorem: Resultant velocity (v) = \(\sqrt{v_{h}^2 + v_{v}^2}\) v = \(\sqrt{(19.3)^2 + (8.9)^2}\)

Calculate the angle using arctangent

To find the angle \(\theta\), we can use the arctangent formula as follows: \(\theta = \arctan(\frac{v_v}{v_h})\) \(\theta = \arctan(\frac{8.9}{19.3})\)

Convert the angle from radians to degrees

Now we have the angle in radians and we need to convert it into degrees. To do this, use the conversion factor: \(\frac{180^{\circ}}{\pi}\) radians = 1 degree \(\theta (degrees) = \theta (radians) * \frac{180^{\circ}}{\pi} \)

Calculate the final angle

Finally, by plugging the values we have, we will obtain the angle \(\theta (degrees) = \arctan(\frac{8.9}{19.3}) * \frac{180^{\circ}}{\pi} \) Now, we can use a calculator or any other mathematical tool to get the final answer for the angle \(\theta\).

Key concepts

Rain's Relative Velocity

When we observe rain from a moving car, its apparent path differs from what we would see if we were standing still. This is due to relative velocity, a concept in physics that describes how the velocity of one object is perceived from another moving object.

In our example, the rain has two components of motion. Vertically, it falls with a speed of 8.9 m/s, which remains constant whether the car is moving or not. Horizontally, things change because the car's motion creates a horizontal component of the rain's velocity from the driver's perspective. This is equal to the speed of the car (19.3 m/s). The driver sees the rain hitting the car at an angle, rather than straight down. By combining the horizontal and vertical velocities, we can determine this apparent angle.

Pythagorean Theorem in Physics

To determine the resultant velocity, or the total velocity of the rain from the driver's perspective, we employ the Pythagorean theorem. This theorem is not only a fundamental principle in geometry but also an essential tool in physics when it comes to combining velocities at right angles to each other.

The resultant velocity is the hypotenuse of a right triangle, with the car's velocity and the rain's vertical velocity forming the other two sides. Mathematically, the magnitude of the resultant velocity (v) is given as: \[v = \sqrt{v_{h}^2 + v_{v}^2}\]
By plugging in the values for horizontal and vertical components, we get the rain’s total velocity with respect to the car.

Arctangent Function

The arctangent function is a fundamental trigonometric function, also known as the inverse tangent. It's invaluable for finding angles when we know the opposite and adjacent sides of a right triangle. In physics, this function often comes into play when we're dealing with direction, such as the angle of the rain's path viewed from a moving vehicle.

The arctangent function provides the angle whose tangent is the ratio of the vertical (opposite) component to the horizontal (adjacent) component of velocity. For our car and rain problem, the angle \(\theta\) is calculated as: \[\theta = \arctan(\frac{v_v}{v_h})\]

Converting Radians to Degrees

After calculating the angle using the arctangent function, we find it measured in radians, a standard unit of angular measurement in mathematics and physics. However, most of us are more familiar with degrees when visualizing angles. To convert radians to degrees, we use the conversion factor \[\frac{180^\circ}{\pi}\]
By multiplying the value in radians by this factor, we transform it into a degree measurement which is more intuitive for most. For example, the angle of rainfall \(\theta\) expressed in degrees will be: \[\theta (degrees) = \theta (radians) \times \frac{180^\circ}{\pi}\]
It's worth noting that this step is crucial to make the measures more understandable and relatable in everyday concepts, like the angle of rain hitting a car window.