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 need to break down the problem into two parts: the rock's motion up the roof and its subsequent motion in free fall.
We start by analyzing the rock's motion up the roof using Newton's second law of motion:
1. Determine the normal force experienced by the rock:
The normal force acts perpendicular to the incline and balances the weight of the rock. We can calculate it using the equation:
Normal force = Weight of the rock * cos(θ)
Weight of the rock = mass of the rock * gravitational acceleration
Gravitational acceleration, g = 9.8 m/s^2
θ = 30.5°
2. Calculate the force of kinetic friction:
The force of kinetic friction acts parallel to the incline and opposes the motion of the rock. We can calculate it using the equation:
Force of kinetic friction = coefficient of kinetic friction * normal force
3. Determine the net force acting on the rock:
Net force = applied force - force of kinetic friction
Since the applied force is horizontal and the vertical motion is perpendicular to it, the applied force does no work in the vertical direction. Therefore, the net force acting on the rock is only the force of kinetic friction.
4. Use Newton's second law to find the acceleration of the rock:
Net force = mass of the rock * acceleration
Rearranging the equation, we have:
Acceleration = net force / mass of the rock
5. Use the equations of motion to find the time taken for the rock to reach its peak:
The rock starts from rest (initial speed = 0), and the distance traveled up the roof is 10.0 m.
We can use the equation:
Distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Plug in the values and solve for time.
After determining the time taken to reach the peak, we move on to the rock's motion in free fall. It follows a parabolic trajectory, so we can use kinematic equations to find the maximum height:
6. Use the equation for vertical displacement to find the maximum height:
The equation is:
Maximum height = vertical displacement = (initial velocity * time) + (0.5 * acceleration due to gravity * time^2)
Plug in the values for time and acceleration due to gravity (g), and solve for the maximum height.
By following these steps, you can determine the maximum height the rock reaches above the point where it was kicked.