Conservation of energy
The idea
Energy bookkeeping is the great shortcut of mechanics: in a system where only gravity and other conservative forces act, the total mechanical energy — kinetic plus potential — stays constant. Energy slides between the forms KE = ½mv² and PE = mgh, but the sum at any snapshot equals the sum at any other. You already know energy is transferred and transformed rather than created; conservation turns that idea into an equation you can solve.
The working method is to pick two snapshots of the motion, choose a height reference, and write KE₁ + PE₁ = KE₂ + PE₂. The magic is what you get to ignore: the path between snapshots, the changing speed along the way, even the shape of a curving track. If friction or air resistance does act, energy is not destroyed — it leaves the mechanical ledger as thermal energy, and you account for it by subtracting the work done against friction.
The frequent misconception is that a steeper or longer track changes the final speed. For frictionless motion it cannot: the speed at the bottom depends only on the vertical drop, because gravity's energy transfer cares only about height change. Two slides of the same height deliver the same final speed, just after different ride times.
Worked example
A 0.50 kg cart is released from rest at the top of a smooth, frictionless track 5.0 m above the ground. Find its speed at the bottom of the track and its speed at a point 1.8 m above the ground.
- Choose the ground as the height reference and write the energy ledger at release: KE = 0 (released from rest) and PE = mgh = 0.50 × 9.8 × 5.0 = 24.5 J, so total mechanical energy is 24.5 J throughout.
- At the bottom all of that energy is kinetic: ½mv² = mgh. The mass cancels, leaving v = √(2gh) = √(2 × 9.8 × 5.0) = √98 ≈ 9.9 m/s.
- At the 1.8 m point only the energy from the 3.2 m drop has converted: ½v² = g × (5.0 − 1.8), so v = √(2 × 9.8 × 3.2) = √62.72 ≈ 7.9 m/s.
- Check the books at 1.8 m: PE = 0.50 × 9.8 × 1.8 = 8.82 J and KE = ½ × 0.50 × 62.72 = 15.68 J, which sum to 24.5 J — the ledger balances.
- Notice what was never needed: the track's shape, its steepness, and the cart's mass all dropped out. Only the vertical drops mattered.
Answer. The cart reaches about 9.9 m/s at the bottom and is moving at about 7.9 m/s when it passes the 1.8 m mark.
Check your understanding
- Why does the final speed at the bottom of a frictionless slide depend on the height but not on the slide's shape or length?
- How would you modify the energy equation if friction removed 6 J of energy between the two snapshots you chose?
- What clues in a problem tell you to reach for energy conservation instead of Newton's second law and kinematics?
- If a roller coaster barely makes it over a second hill, what does that tell you about the hill's height compared with the first one?
Build the foundations first
Conservation of energy builds on these concepts. If any feel shaky, start there.