A 2.4 kg box has an initial velocity of 3.6 m/s upward along a plane inclined at 27◦ to the horizontal. The coefficient of kinetic friction between the box and the plane is 0.12. The acceleration of gravity is 9.81 m/s 2. How far up the incline does the box travel? Answer in units of m.

d = 1.18 m

Explanation:

In abscense of friction, total mechanical energy must be constant, i. e.,

ΔK + ΔU = 0

As we are told that there exists a kinetic friction between the box and the plane, we need to take into account the work done by the friction force in the equation, as follows:

ΔK + ΔU = Wnc (1)

If we take as our zero gravitational potential energy reference, the height at which the box is sent upward, we can write the following equations for the different terms in (1):

ΔK = Kf - K₀ = 0 - 1/2*m*v₀² = - 1/2*2.4kg * (3.6) ² (m/s) ² = - 15.6 J

ΔU = Uf - U₀ = m*g*h = * m*g*d*sin θ = 2.4 kg*9.81 m/s²*d*0.454 = 10.7*d J

Wnc = Ff. d * cos (180º) = μk*N*d*cos (180º) (2)

The friction force always opposes to the displacement, so the angle between force and displacement is 180º.

The normal force, as is always perpendicular to the surface, takes the value needed to equilibrate the component of the weight perpendicular to the incline, as follows:

N = m*g*cosθ = 2.4 kg*9.81 m/s²*cos 27º = 21 N

Replacing in (2):

Wnc = 0.12*21*cos (180º) = - 2.52*d J

Replacing in (1):

-15.6 J + 10.7*d J = - 2.52*d J

Solving for d:

d = 1.18 m