Release a cart at the top of a slope. It speeds up because gravity is pulling it, but the rail is pushing on it too. The first thing to do is separate the two.
The rail always pushes perpendicular to its surface. The cart travels along that surface, so the two directions meet at a right angle.
A force does work only when it has a component along the direction of motion. The normal force has no component along the direction of travel. So however hard it pushes, the work it does is .
That leaves gravity as the only thing changing the speed. The rail alters the direction of the cart and leaves its speed alone. Everything that follows rests on this.
The work done by gravity can be rewritten as a loss of potential energy. At a height the potential energy is .
Where the height is measured from may be chosen freely, so long as the choice is fixed. The value of itself depends on that choice, but only differences are ever used. The figure takes the level section as .
What matters is that is fixed by the height alone. Not where the cart is on the course, only how high it is. At the same height the value is the same, halfway up a slope or beside a loop.
The path does not appear in it. That is the subject of this lecture.
The kinetic energy is . The sum of this and is called the mechanical energy.
Gravity was the only thing doing work, and the work gravity does is precisely the loss in . So falls by as much as rises, and the sum does not change.
The bar in the figure is that sum. The dividing line moves; the full length does not. Run down a slope and the line shifts to give more ; climb a hill and it shifts back.
As an equation, . At the point of release , so only is left on the right.
Solve that equation for , which gives .
The mass has gone. A heavy cart and a light one released from the same height reach the same speed.
The path has gone as well. Steep slope or gentle, descend to the same height and the speed is the same. The two carts in the figure run down differently shaped courses, yet while the dashed line marks the same height the arrows stay the same length.
What does differ is how long it takes to get there. The steeper slope arrives sooner. Speed is fixed by height, time by the path. Keep the two apart.