A car is driving along at 16 m/s, and the rain is falling at 6 m/s straight down. What angle θ (in degrees) does the rain make with respect to the horizontal as observed by the driver?
tan A = Y/X = 6/16 = 0.375
A = 20.56o
To find the angle θ that the rain makes with respect to the horizontal as observed by the driver, we can use trigonometry.
Let's assume that the car is moving to the right, in the positive x-direction. The rain is falling straight down, along the negative y-direction.
First, we need to calculate the relative velocity of the rain with respect to the car. This can be found by subtracting the velocity of the car from the velocity of the rain. The resulting velocity vector points at an angle θ with respect to the horizontal.
Using vector subtraction, we can find the relative velocity of the rain:
Relative velocity = Rain velocity - Car velocity
In magnitude, it is given by:
Relative velocity = √((6 m/s)^2 + (16 m/s)^2)
Next, we can find the angle θ using the inverse tangent function:
θ = tan^(-1)(y-component of relative velocity / x-component of relative velocity)
In this case, the y-component of the relative velocity is the rate at which rain is falling (6 m/s), and the x-component is the speed of the car (16 m/s).
Substituting these values, we have:
θ = tan^(-1)(6 m/s / 16 m/s)
Evaluating this expression gives us the value of θ.
Let's solve this:
θ = tan^(-1)(6/16) ≈ 20.56 degrees
Therefore, the angle θ that the rain makes with respect to the horizontal as observed by the driver is approximately 20.56 degrees.
To find the angle θ, we need to consider the relative motion between the car and the rain. Let's break down the problem step by step:
Step 1: Understanding the scenario
The car is moving horizontally with a velocity of 16 m/s, while the rain is falling vertically with a speed of 6 m/s. We want to find the angle at which the rain appears to be falling from the perspective of the driver inside the moving car.
Step 2: Analyzing the components of motion
Let's consider the horizontal and vertical components of motion:
- Vertical component: The rain is falling straight down with a velocity of 6 m/s.
- Horizontal component: The car is moving horizontally with a velocity of 16 m/s.
Step 3: Calculating the angle
To determine the angle θ, we can calculate the tangent of the angle using the vertical and horizontal components of motion:
tan(θ) = vertical component / horizontal component
Since the vertical component is the same as the actual speed of the rain (6 m/s), and the horizontal component represents the velocity of the car (16 m/s), we have:
tan(θ) = 6 m/s / 16 m/s
Step 4: Solving for θ
To find the angle θ, we need to take the inverse tangent (arctan) of both sides:
θ = arctan(6 m/s / 16 m/s)
Now, we can use a scientific calculator or an online calculator to find the arctan value:
θ ≈ 20.565°
Therefore, the rain appears to be falling at an angle of approximately 20.565 degrees with respect to the horizontal as observed by the driver.