One side of the roof of a building slopes up at 30.5°. A roofer kicks a round, flat rock that has been thrown onto the roof by a neighborhood child. The rock slides straight up the incline with an initial speed of 15.0 m/s. The coefficient of kinetic friction between the rock and the roof is 0.370. The rock slides 10.0 m up the roof to its peak. It crosses the ridge and goes into free fall, following a parabolic trajectory above the far side of the roof, with negligible air resistance. Determine the maximum height the rock reaches above the point where it was kicked.
To determine the maximum height the rock reaches above the point where it was kicked, we can break down the problem into two parts: the rock's motion up the roof and its subsequent free fall.
1. Motion up the roof:
The first step is to find the normal force, gravitational force, and frictional force acting on the rock as it slides up the roof.
- Normal force (N): The normal force is equal to the component of the rock's weight perpendicular to the roof. N = mg * cos(θ), where m is the mass of the rock, g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of inclination (30.5° in this case).
- Gravitational force (Fg): The gravitational force is equal to the weight of the rock. Fg = mg, where m is the mass of the rock.
- Frictional force (Ff): The frictional force opposes the motion of the rock up the roof. Ff = μN, where μ is the coefficient of kinetic friction (0.370) and N is the normal force.
Since the rock slides up the roof with constant velocity, the frictional force is equal in magnitude and opposite in direction to the component of the gravitational force parallel to the roof. Therefore, Ff = Fg * sin(θ).
2. Free fall:
Once the rock crosses the ridge, it goes into free fall. We can use the kinematic equations to determine the maximum height it reaches above the point where it was kicked.
- Initial velocity (vi): The initial velocity of the rock at the peak of the roof is the final velocity while sliding up the roof.
- Final velocity (vf): The final velocity at the peak is zero since the rock momentarily comes to rest before falling.
- Acceleration (a): The acceleration due to gravity acts vertically downward and remains constant during free fall. a = -g, where g is the acceleration due to gravity. The negative sign indicates that the acceleration is in the opposite direction of the positive vertical axis.
- Displacement (y): The maximum height reached by the rock above the point where it was kicked.
Using the following kinematic equation for vertical motion:
vf^2 = vi^2 + 2ad,
we can rearrange it to solve for the maximum height (y).
3. Steps to calculate the maximum height:
1. Calculate the normal force (N) using the formula N = mg * cos(θ).
2. Calculate the gravitational force (Fg) using the formula Fg = mg.
3. Calculate the frictional force (Ff) using the formula Ff = Fg * sin(θ).
4. Calculate the initial velocity (vi) using the given initial speed (15.0 m/s).
5. Calculate the maximum height (y) using the kinematic equation vf^2 = vi^2 + 2ad, where vf = 0, vi is the initial velocity, a is the acceleration due to gravity, and d is the displacement.
By following these steps and plugging in the given values, you can determine the maximum height the rock reaches above the point where it was kicked.