$1.$ A toy train has a non$-$zero velocity $0$ at the top of a hill. The train travels down the hill, then along a flat horizontal surface until it collides with an elastic bumper that brings the train to a stop. Assume the hill and flat surface are frictionless and neglect air resistance.

The same toy train setup is used, but now there is friction along the horizontal surface $(2)$ with a coefficient of kinetic friction equal to $0\mathrm{.}3.$ The distance $2$ is $40.$

i$.$ How much does friction cause the thermal energy of the system to increase as the train travels along the horizontal surface $(2)?$

j$.$ Calculate the new distance the spring is compressed when the train comes to a stop.

k$.$ Suppose the coefficient of kinetic friction was equal to $0\mathrm{.}5.$ How much would the thermal energy increase as the train travels along the horizontal surface $(2)?$

l$.$ How does your answer to part k compare to the initial mechanical energy of the train at the top of the hill? What does this imply about the motion of the train?

m$.$ Using this larger coefficient of friction, determine how far along the horizontal surface the train travels before coming to a stop.