One side of the roof of a building slopes up at 37.5°. 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 0.395. 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.

maximum height  above point kicked  8.2 m

Explanation:

With we have a friction force, let's use Newton's second law to find the acceleration of the body, let's look at the diagram in the attacment

                 -fr -W sin θ = m a

                 N - W cos θ = 0       ⇒      N = mg cos θ

The equation for the force of friction is fr = μN, substitute and calculate

               -μ (mg cos θ) - mg sin θ = ma

               -μ  g cos θ-  g sin θ = a

                a = -g (μ cos θ + sin θ)

                a = - 9.8 (0.395 cos 37.5 + sin 37.5)

                a = - 9.04 m/s²

The negative sign indicates that the acceleration is stopping the movement.

With this acceleration we can calculate the speed of the stone when it reaches the end of the roof

                   Vf² = Vo² - 2 to Y

The distance of 10 m is along the ceiling, let's calculate with trigonometry when the height rises

                       sin θ=Y / L     ⇒    Y = L sin θ

                       Y = 10.0 sin 37.5

                       Y = 6 m

Let's replace and calculate the speed

                      Vf = √(15 2 - 2 9.04 6)

                      Vf = 10.8 m / s

This is the speed with which the stone reaches the end of the roof, from where its parabolic movement begins, where is the initial velocity Vo = 10.8 m/s with the roof angle T = 37.5º. Let's calculate the maximum height reached at this point the vertical speed must be zero (Vy = 0)

Let's look for trigonometry the components of the initial velocity

                   Vox = Vo cos θ

                   Voy = Vo sin Tθ

                   Vox = 10.8 cos 37.5

                   Voy = 10.8 sin 37.5

                   Vox = 8.33 m / s

                   Voy = 6.57 m / s

Let's calculate the maximum height

                   Vyf² = Voy² - 2 g Y

                   0 = Voy2 - 2 g Y

                   Y = Voy² / 2g

                   Y = 6.57²/2 9.8

                   Y = 2.2 m

This is the maximum height from the tip of the roof

Measured from the point where the stone was kicked we must add the height of advanced on the roof

                  Y = Y roof + Y max

                  Y = 6 + 2.2

                  Y = 8.2 m

This is the height measured from the point where it was kicked

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