A car is parked on a cliff overlooking the ocean on an incline that makes an angle of 25.0° below the horizontal. The negligent driver leaves the car in neutral, and the emergency brakes are defective. The car rolls from rest down the incline with a constant acceleration of 3.97 m/s2 for a distance of 50.0 m to the edge of the cliff, which is 40.0 m above the ocean.
(a) Find the car's position relative to the base of the cliff when the car lands in the ocean.
m
(b) Find the length of time the car is in the air.
s
To find the car's position relative to the base of the cliff when it lands in the ocean, we can break down the motion of the car into two separate components: horizontal and vertical.
First, let's consider the horizontal motion. The car rolls down the incline with a constant acceleration of 3.97 m/s^2. We can use the kinematic equation:
d = v0 * t + (1/2) * a * t^2
Where:
d = horizontal distance traveled (50.0 m)
v0 = initial horizontal velocity (0 m/s)
a = horizontal acceleration (3.97 m/s^2)
t = time
Since the car starts from rest (v0 = 0), the equation simplifies to:
d = (1/2) * a * t^2
Rearranging the equation, we can solve for time:
t = sqrt((2 * d) / a)
Now, substitute the given values:
t = sqrt((2 * 50.0 m) / 3.97 m/s^2)
t ≈ 5.03 s
So, it takes approximately 5.03 seconds for the car to reach the edge of the cliff.
Now, let's consider the vertical motion. The car is initially at a height of 40.0 m. We can use the kinematic equation:
d = v0 * t + (1/2) * g * t^2
Where:
d = vertical distance traveled (40.0 m)
v0 = initial vertical velocity (0 m/s)
g = acceleration due to gravity (-9.8 m/s^2)
t = time
Since the car starts from rest (v0 = 0), the equation simplifies to:
d = (1/2) * g * t^2
Rearranging the equation, we can solve for time:
t = sqrt((2 * d) / g)
Substitute the given values:
t = sqrt((2 * 40.0 m) / 9.8 m/s^2)
t ≈ 2.04 s
So, it takes approximately 2.04 seconds for the car to reach the ocean after leaving the cliff.
(a) The car's position relative to the base of the cliff when it lands in the ocean can be found by multiplying the horizontal velocity (which can be obtained using the horizontal acceleration and time) by the time it takes to reach the ocean:
distance = horizontal velocity * time = (acceleration * time) * time = (3.97 m/s^2 * 5.03 s) * 5.03 s = 99.92991 m
Therefore, the car's position relative to the base of the cliff when it lands in the ocean is approximately 99.93 m.
(b) The length of time the car is in the air can be calculated by subtracting the time it takes for the car to reach the surface of the ocean (2.04 s) from the total time it takes for the car to reach the edge of the cliff (5.03 s):
air time = total time - time to reach the ocean = 5.03 s - 2.04 s = 2.99 s
Therefore, the car is in the air for approximately 2.99 seconds.