You and your friend peter are putting new shingles on a roof pitched at 20°. you're sitting on the very top of the roof when peter, who is on the edge of the roof, 5.1 m away, asks you for the box of nails. rather than carry the 2.2 kg box of nails down to peter, you decide to give the box a push and have it slide down to him.
if the coefficient of kinetic friction between the box and the roof is 0.53, with what speed should you push the box to have it gently come to rest right at the edge of the roof?

vi = 3.95 m/s

Explanation:

We can apply the Work-Energy Theorem as follows:

W = ΔE = Ef - Ei

W = - Ff*d

then

Ef - Ei = - Ff*d  

If

Ei = Ki + Ui = 0.5*m*vi² + m*g*hi = 0.5*m*vi² + m*g*hi = m*(0.5*vi² + g*hi)

hi = d*Sin 20º = 5.1 m * Sin 20º = 1.7443 m

Ef = Kf + Uf = 0 + 0 = 0

As we know,   vf = 0  ⇒ Kf = 0

Uf = 0  since hf = 0

we get

W = ΔE = Ef - Ei = 0 - m*(0.5*vi² + g*hi)  ⇒   W = - m*(0.5*vi² + g*hi)   (I)

If

W = - Ff*d = - μ*N*d = - μ*(m*g*Cos 20º)*d = -  μ*m*g*Cos 20º*d    (II)

we can say that

- m*(0.5*vi² + g*hi) = -  μ*m*g*Cos 20º*d  

⇒ vi = √(2*g*(μ*Cos 20º*d - hi))  

⇒ vi = √(2*(9.81 m/s2)*(0.53*Cos 20º*5.1m - 1.7443 m)) = 3.95 m/s

lol

explanation: