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 66.87° from the vertical.

Step-by-step solution

Convert car speed to m/s

To convert the speed of the car from km/h to m/s, we will use the conversion rate that 1 km/h equals to 0.2778 m/s. \(60.0 \mathrm{~km/h} \times 0.2778 (\mathrm{m/s}) / (\mathrm{km/h}) = 16.7 \mathrm{~m/s}\)

Find the tangent of the angle

In the right triangle formed, the opposite side is the horizontal speed of the car, \(16.7 \mathrm{~m/s}\), and the adjacent side is the vertical speed of the raindrops, \(7.00 \mathrm{~m/s}\). Using the definition of tangent, we have: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{16.7}{7.00}\)

Find the angle using inverse tangent

To find the angle, we will use the inverse tangent function: \(\theta = \tan^{-1}\left(\frac{16.7}{7.00}\right) = 66.87^{\circ}\) So, to the driver of the car, the raindrops appear to be falling at an angle of \(66.87^{\circ}\) from the vertical.

Key concepts

Frame of Reference

Understanding the concept of frame of reference is crucial when analyzing motion in physics. Imagine you're at a still point on the sidewalk, watching rain fall straight down. To you, standing still, the rain appears to fall vertically. However, if you're moving, such as driving a car, raindrops may seem to come at an angle. A frame of reference is, in simple terms, the viewpoint from which you observe motion. It's like choosing the vantage point for a camera to capture a scene. When the driver moves at a certain speed, the velocity of the rain relative to the driver changes, which alters the observed motion of the raindrops. Consequently, calculating relative velocity requires taking the frame of reference into account, affecting how we perceive motion and orientation.

Velocity Conversion

The process of velocity conversion is a practical skill in physics, specifically when dealing with problems involving relative velocity. Since different regions use different units for speed, converting units becomes a daily activity for scientists and engineers. In the given problem, the car's speed is provided in kilometers per hour (km/h), but for the purposes of solving physics problems, it's often more convenient to work in meters per second (m/s). This conversion is done by knowing the equivalence: 1 km/h is equal to 0.2778 m/s. By multiplying the car's speed by this conversion factor, we align the units of both velocities (car's speed and rain's speed) to be in the same system, allowing for accurate calculations within the selected frame of reference.

Tangent Function

The tangent function plays a vital role in trigonometry, especially when it comes to right-angled triangles. It is defined as the ratio of the length of the opposite side to that of the adjacent side of a given angle in a right triangle. Specifically, for an angle \( \theta \), it is represented as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). In our exercise, the car's horizontal speed acts as the 'opposite side,' while the vertical speed of the rain serves as the 'adjacent side.' When we apply the tangent function to the velocity components, it helps us determine the angle at which the motion of the rain is observed relative to the driver. Grasping the use of the tangent function is pivotal as it links the geometry of the situation to the numerical values of velocity.

Inverse Trigonometric Functions

To find an angle when you know the sides of a right triangle, you use inverse trigonometric functions. These functions are the inverses of the regular trigonometric functions like sine, cosine, and tangent. They allow you to work backwards from the ratio of sides to find the angle that produced them. For example, the inverse tangent (also known as arctan or \( \tan^{-1} \)) is used to find the angle whose tangent is a given number. By inputting the ratio calculated from the sides of a triangle, the inverse tangent function gives you the measure of the angle in either degrees or radians. This is exactly what is done in the final step of our exercise: after finding the tangent of the angle, we apply the inverse tangent to get the actual angle of the rain's apparent trajectory from the perspective of the moving driver.