Problem: A 5.3 kg body is held initially at rest against a spring which has been compressed 20 cm (k = 300 N/m). When it is released, it slides up the rough incline (kinetic coefficient = 0.153). When it lands (at a point 25 cm below its starting point), what is its speed?

Solution: The mass is in contact with the spring for 20 cm and acquires a speed as a result of the decompression. After this, I am assuming that it moves fast enough despite friction so that the spring does not continue to exert a force on it. It slides on the incline, losing speed until it reaches the end of the incline. Then it becomes a projectile, assuming that it still has some speed.

If the surface were smooth, I could merely look at the energy before and the energy after. But the energy lost to friction has to be considered. Taking the reference level for gravitational potential energy to be the ground level,

The initial height is 25 cm; c is the amount the spring is compressed (20 cm). W_f is the work done against friction (the heat produced). I'll need a free body diagram to get N, and s is the distance moved on the incline (1.24 m).

Returning to the energy equation...

You might like to test some of the assumptions made? Are they valid?