First, consider the case without friction. A small object released from rest on a fixed, smooth rail passes through the bottom and climbs the slope on the opposite side.
Adjust conditions:
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高さ1.0 mから放すと、反対側ではどの高さまで上がるか？
底に長さ1.0 mの粗い区間（動摩擦係数0.20）があると、反対側の最高点は？
The equations explain the phenomenon
1The derivation begins with the equation of motionmdvdt=F……
2Both sides are multiplied by the velocity.mvdvdt=Fv……
3The left-hand side is expressed as a time derivative.d(½mv²)dt=Fv……
4Velocity is expressed as the time derivative of position.d(½mv²)dt=Fdxdt……
5Force is expressed as a derivative of potential energy.d(½mv²)dt=−(dUdx)dxdt……
6The right-hand side is expressed as a time derivative.d(½mv²)dt=−dUdt……
7Adding dU/dt to both sides combines the terms into the time derivative of the total energy.d(½mv² + U)dt=0……
8We identify the quantity that remains unchanged with time.½mv²+U=Constant……
9The condition for conservation is applied to motion along the rail.m dvdt=−m g sinθ=−d(m g y)ds……
10Conservation of mechanical energy equates the energy at release to the energy at the highest point.ymax=h₀……
With friction: calculate the decrease in mechanical energy.
11The decrease in energy is calculated from the work done by friction.ymax=h₀−μ L……
What this chapter has shown
When the force can be written as F=−dU/dx using a U with no explicit time dependence, the equation of motion gives d(½mv²+U)/dt=0. Kinetic and potential energy may change, but their sum remains constant.
1.00 mHeight of the highest point on the opposite side
With friction along a 1.0 m section at the bottom (μ=0.20), the object reaches a height of 0.80 m.
In everyday lifeFor a roller coaster with no additional power input during the ride, subsequent hills are lower than the first hill to account for the loss of mechanical energy due to friction and air resistance.
Detailed summary
The equation ½mv² + U=constant means that even as kinetic energy and potential energy change, their sum remains constant over time. Deriving this result from the equation of motion shows that mechanical energy is conserved because the increase in one form of energy equals the decrease in the other. The derivation assumes that the force depends only on position. Understanding the derivation allows you to determine for yourself whether the same formula applies when friction is present.