A car is parked on a cliff overlooking the ocean on an incline that makes an angle of 20.0° 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 3.21 m/s2 for a distance of 35.0 m to the edge of the cliff, which is 30.0 m above the ocean.
(a) Find the car's position relative to the base of the cliff when the car lands in the ocean.
(b) Find the length of time the car is in the air.
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To solve this problem, we can use the principles of motion and apply them to the situation described. We'll divide the problem into two parts: the motion along the incline and the projectile motion when the car leaves the incline and lands in the ocean.
(a) Finding the car's position relative to the base of the cliff when it lands in the ocean:
1. Determine the time it takes for the car to reach the edge of the cliff.
We can use the kinematic equation:
d = v₀t + (1/2)at²
where d is the distance traveled along the incline (35.0 m), v₀ is the initial velocity (0 m/s, since the car starts from rest), a is the acceleration along the incline (3.21 m/s²), and t is the time taken. Rearranging the equation, we get:
t = √(2d/a)
Plugging in the values, we have:
t = √(2 * 35.0 m / 3.21 m/s²)
2. Find the horizontal distance covered by the car.
Using trigonometry, we can determine the horizontal component of the car's motion during the time it reaches the edge of the cliff.
Horizontal distance = d * cos(angle), where angle is the angle of the incline (20.0°).
3. Calculate the position relative to the base of the cliff when the car lands.
Position relative to the base of the cliff = horizontal distance - distance of the cliff from the top of the incline (30.0 m).
(b) Finding the length of time the car is in the air:
1. Determine the vertical displacement of the car when it falls off the cliff.
The vertical displacement can be calculated using the equation:
d = v₀t + (1/2)at²
where d is the vertical displacement (30.0 m), v₀ is the initial vertical velocity (0 m/s, since the car falls from rest), a is the acceleration due to gravity (9.8 m/s²), and t is the time taken.
2. Find the time taken for the car to reach the maximum height during its projectile motion.
At the highest point of the trajectory, the vertical velocity becomes zero before it starts falling back down. We can use the equation:
v = v₀ + at
where v is the vertical velocity (0 m/s), v₀ is the initial vertical velocity, a is the acceleration due to gravity (-9.8 m/s²), and t is the time taken. Rearranging the equation, we can solve for t.
3. The total time the car is in the air is double the time calculated in step 2.
By following these steps, you can find the car's position relative to the base of the cliff when it lands in the ocean (a) and the length of time the car is in the air (b).