Your friend's car is parked on a cliff overlooking the ocean on an incline that makes an angle of 16.9° below the horizontal. The brakes fail, and the car rolls from rest down the incline for a distance of 36.3 m to the edge of the cliff, which is 61.1 m above the ocean, and, unfortunately, continues over the edge and lands in the ocean.

a) Find the car's position relative to the base of the cliff when the car lands in the ocean.

b) Find the length of time the car is in the air.

Let's solve the problem step by step:

Step 1: Find the horizontal distance covered by the car.
Given:
Angle of the incline (θ) = 16.9°
Distance rolled down the incline (x) = 36.3 m

To find the horizontal distance covered by the car, we can use trigonometry. Since the angle given is below the horizontal, we can use the cosine function:
Horizontal distance (d) = x * cos(θ)

Substituting the values:
d = 36.3 m * cos(16.9°)
d ≈ 34.69 m

So, the car covers a horizontal distance of approximately 34.69 m.

Step 2: Find the vertical distance the car falls.
Given:
Height of the cliff (h) = 61.1 m

The car falls vertically from the top of the cliff to the ocean. Therefore, the vertical distance fallen is equal to the height of the cliff.

Vertical distance fallen (y) = h = 61.1 m

So, the car falls a vertical distance of 61.1 m.

Step 3: Find the car's position relative to the base of the cliff when it lands in the ocean.
To find the car's position, we can use the Pythagorean theorem since the horizontal and vertical distances form a right triangle.

Using the formula:
Distance^2 = Horizontal distance^2 + Vertical distance^2

Distance^2 = (34.69 m)^2 + (61.1 m)^2
Distance^2 ≈ 1207.0361 m^2

Taking the square root of both sides:
Distance ≈ √(1207.0361 m^2)
Distance ≈ 34.76 m

Therefore, the car's position relative to the base of the cliff when it lands in the ocean is approximately 34.76 meters.

Step 4: Find the length of time the car is in the air.
To find the time, we can use the equation of motion:

y = ut + (1/2)at^2

Where:
y = vertical distance fallen (61.1 m)
u = initial vertical velocity (0 m/s, as the car starts from rest)
a = acceleration due to gravity (-9.8 m/s^2, assuming no air resistance)
t = time

Substituting the values:
61.1 = 0*t + (1/2)*(-9.8)*t^2

Rearranging the equation:
4.9t^2 = 61.1

Simplifying:
t^2 = 61.1/4.9
t^2 ≈ 12.5

Taking the square root of both sides:
t ≈ √12.5
t ≈ 3.54 s

Therefore, the car is in the air for approximately 3.54 seconds.

To find the car's position relative to the base of the cliff when it lands in the ocean, we need to break down the motion into horizontal and vertical components.

a) Horizontal motion: The horizontal component of the car's motion is not affected by the incline or the brakes failing. Therefore, the car travels a horizontal distance of 36.3 m.

Vertical motion: To find the vertical distance traveled by the car, we need to consider the height of the cliff and the angle of the incline.

Using the given information, we can form a right triangle with the height of the cliff (61.1 m) as the vertical leg and the distance traveled horizontally (36.3 m) as the horizontal leg. The angle between the incline and the horizontal is 16.9°.

Using trigonometry, we can find the vertical distance traveled:

Vertical distance = Horizontal distance * tan(angle)
Vertical distance = 36.3 m * tan(16.9°)

Now, let's calculate this value:

Vertical distance = 36.3 m * tan(16.9°)
Vertical distance ≈ 10.14 m

Therefore, the car's position relative to the base of the cliff when it lands in the ocean is approximately 10.14 meters below the top of the cliff.

b) To find the length of time the car is in the air, we can use the vertical motion equation:

Vertical distance = (Initial vertical velocity * time) + (0.5 * acceleration * time^2)

Since the car starts from rest vertically (no initial vertical velocity) and falls due to gravity (acceleration = 9.8 m/s^2), the equation simplifies to:

Vertical distance = 0.5 * acceleration * time^2

Plugging in the values we know (vertical distance = 61.1 m, acceleration = 9.8 m/s^2), we can solve for time:

61.1 m = 0.5 * 9.8 m/s^2 * time^2

Rearranging the equation, we have:

time^2 = (2 * 61.1 m) / 9.8 m/s^2
time^2 ≈ 12.449 s^2

Taking the square root of both sides, we find:

time ≈ 3.53 seconds

Therefore, the length of time the car is in the air is approximately 3.53 seconds.