a 2kg block is pushed with a force of 15 N across a horizontal surface. The coefficient of kinetic friction between the block and the surface is 0.2. The block starts from rest and is pushed for a distance of 10 meters. After being pushed, the block moves up a frictionless ramp until it comes to rest. How high does the block travel up the ramp?

According to the work-energy theorem, the work (W) done on the block is equal to the change in its mechanical energy (ΔE), where the mechanical energy is the sum of its kinetic energy (KE) and gravitational potential energy (PE). The work done is equal to the net force (∑F) times the parallel distance (d). The kinetic energy is half the mass (m) times the square of the speed (v), and the potential energy is the weight (mg) times the height (h).

To find the net force, we'll draw a free body diagram, which shows all the forces acting on the block. In this case, there are four forces:

Weight force mg pulling down
Normal force N pushing up
Friction force Nμ pushing left, where μ is the coefficient of friction
Applied force F pushing right

The block moves horizontally, so the net force in the vertical direction is zero.

N − mg = 0

N = mg

The net force in the horizontal direction is:

∑F = F − Nμ

∑F = F − mgμ

The work done on the block is:

W = ∑F d

W = (F − mgμ) d

The block starts and stops at rest, so the kinetic energy is zero at both places. Therefore, the work done is equal to the change in potential energy.

W = ΔPE

W = mgΔh

(F − mgμ) d = mgΔh

Δh = (F − mgμ) d / (mg)

Plugging in values:

Δh = [ 15 N − (2 kg) (9.8 m/s²) (0.2) ] (10 m) / [ (2 kg) (9.8 m/s²) ]

Δh = 5.65 m