Question

you have a summer job with an insurance company and have been asked to help with the investigation of a tragic "accident." When you visit the scene, you see a road running straight down a hill which has a slope of 12 degrees to the horizontal. At the bottom of the hill, the road goes horizontally for a very short distance becoming a parking lot overlooking a cliff. The cliff has a vertical drop of 453 feet to the horizontal ground below where a car is wrecked 33 feet from the base of the cliff. Was it possible that the driver fell asleep at the wheel and simply drove over the cliff? After looking pensive, your boss tells you to calculate the speed of the car (in mph) as it left the top of the cliff.

Answer 1

To determine if it was possible for the driver to fall asleep at the wheel and drive over the cliff, we can calculate the speed of the car as it left the top of the cliff using the principles of projectile motion.

Let's break down the problem:

1. The car starts from a horizontal road on a hill with a 12-degree slope to the horizontal.
2. It then goes horizontally for a short distance before going over a cliff with a vertical drop of 453 feet.
3. The wrecked car is 33 feet from the base of the cliff.

First, we need to find the horizontal distance the car traveled on the slope before going over the cliff. This distance can be calculated using trigonometry:

Horizontal distance on the slope = (453 feet) * tan(12 degrees) ≈ 90.35 feet

Now, we know that the car traveled 90.35 feet horizontally before going over the cliff. The remaining horizontal distance between the car's position and the base of the cliff is 33 feet.

So, the total horizontal distance the car traveled before going over the cliff is 90.35 feet + 33 feet ≈ 123.35 feet.

Now, we can calculate the time it took for the car to fall vertically 453 feet using the equations of motion under gravity:

h = (1/2) * g * t^2

Where:
h = vertical distance (453 feet)
g = acceleration due to gravity (32.2 feet/s^2, approximately)

Now, solve for time (t):

453 = (1/2) * 32.2 * t^2
t^2 = (2 * 453) / 32.2
t^2 ≈ 28.11
t ≈ √28.11 ≈ 5.31 seconds

Now that we know the time it took for the car to fall, we can calculate its initial horizontal velocity (speed) when it left the top of the cliff:

Horizontal distance = initial horizontal velocity * time
123.35 feet = initial horizontal velocity * 5.31 seconds

Now, let's convert the units to miles and hours:

1 mile = 5280 feet
1 hour = 3600 seconds

Initial horizontal velocity ≈ (123.35 feet / 5280 feet per mile) / (5.31 seconds / 3600 seconds per hour)

Initial horizontal velocity ≈ 0.036 miles per hour (mph)

So, the speed of the car as it left the top of the cliff was approximately 0.036 mph. This is an extremely low speed, and it's highly unlikely that the driver fell asleep at the wheel and simply drove over the cliff. Other factors or events may have contributed to the car going over the cliff.