AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/3/3-4)
Unit 3 · Topic 3.4
3.4 Conservation of Energy
Energy is never created or destroyed. Whether a system's total energy stays constant depends on how you choose the system: if no outside work is done on it, its energy is constant, and if no friction-like forces act inside it, its mechanical energy (kinetic plus potential) is constant too. Energy bar charts make this bookkeeping visible.
Key terms
- conservation of energy
- mechanical energy
- choice of system
- energy bar chart
- thermal energy
- nonconservative force
Energy is always conserved
Energy can change form or move between a system and its surroundings, but the total is always conserved. If one kind of energy in a system goes up or down, something makes up the difference: another kind inside the system changes the other way, or energy flows across the system's boundary.
In equation form: ΔE_system = W_external. If the outside world does positive work on the system, the system's energy goes up by that much. If the system does work on its surroundings, its energy goes down.
Mechanical energy
Mechanical energy is the sum of a system's kinetic and potential energies: E = K + U. It stays constant when two conditions hold: no external work is done on the system, and no nonconservative forces (like friction) act between its parts. Then K_i + U_i = K_f + U_f.
When a ball falls in the ball–Earth system, U_g drops and K rises by exactly the same amount. When a spring launches a block on a frictionless track, U_s turns into K.
Choosing the system
The same situation can be described with different systems, and the bookkeeping changes:
| System | What energy it holds | How gravity appears |
|---|---|---|
| Ball alone | kinetic only | an external force that does work |
| Ball + Earth | kinetic and gravitational potential | an internal force; no work term |
| Block + spring + Earth | kinetic, elastic and gravitational potential | an internal force; the spring force is internal too |
Friction and thermal energy
Nonconservative forces such as friction and air resistance transform mechanical energy into thermal energy (the surfaces warm up) and sound. If the sliding surfaces are inside your system, the total energy is still constant, but mechanical energy decreases: K_i + U_i = K_f + U_f + ΔE_thermal, where ΔE_thermal = F_f d for kinetic friction over a path of length d.
AP Physics 1 only asks you to track mechanical energy and to recognize that friction turns it into thermal energy or sound. How thermal energy moves by heating is AP Physics 2.
Energy bar charts
An energy bar chart shows each type of energy in the system at the start and at the end as bars, plus a bar for any work done by outside forces in between. The bars must balance: initial energies + external work = final energies. Draw bars to scale when you know the values, and draw no bar (not a tiny one) for an energy that's zero.
Bar charts are a quick check on a solution. If your chart shows energy appearing from nowhere, your system choice or your equation needs fixing.
A problem-solving plan
- Choose the system and say which objects are in it.
- Pick the initial and final moments, and the zero of U_g.
- List each energy at both moments and any work done from outside.
- Write E_initial + W_external = E_final, solve in symbols, then substitute numbers.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Frictionless slide
A child starts from rest at the top of a frictionless slide 20 m above the ground. Use g = 9.8 m/s². How fast is she going when she's 5.0 m above the ground?
Show the solutionHide the solution
- Step 1: System: child + Earth. No external work and no friction, so mechanical energy is constant.
- Step 2: K_i + U_i = K_f + U_f: 0 + mg(20) = ½mv² + mg(5.0).
- Step 3: Mass cancels: v = √[2g(20 − 5.0)] = √(2 × 9.8 × 15) ≈ 17 m/s.
- Step 4: The answer doesn't depend on her mass or on the slide's shape, only on the 15 m drop.
Answer: About 17 m/s
- Example 2Calculator allowed
Spring launcher (classic trap)
A toy launcher has a spring with k = 800 N/m compressed 0.10 m. It fires a 0.20 kg ball straight up. Use g = 9.8 m/s² and ignore air resistance. How high above its starting (compressed) position does the ball rise?
Show the solutionHide the solution
- Step 1: System: ball + spring + Earth. Initially all the energy is elastic: U_s = ½(800)(0.10)² = 4.0 J.
- Step 2: At the top the ball is momentarily at rest, so all the energy is gravitational: mgh = 4.0 J.
- Step 3: h = 4.0 ÷ (0.20 × 9.8) ≈ 2.0 m above the starting position.
- Step 4: The trap is measuring h from the point where the ball leaves the spring. The ball rises 0.10 m while the spring is still pushing, so it ends about 1.9 m above the release point.
Answer: About 2.0 m above its starting position
- Example 3Calculator allowed
Energy lost to friction on a ramp
A 2.0 kg block starts from rest at the top of a ramp 3.0 m high and reaches the bottom at 6.0 m/s. The path along the ramp is 5.0 m long. Use g = 9.8 m/s². How much mechanical energy became thermal energy, and what was the average friction force?
Show the solutionHide the solution
- Step 1: System: block + Earth + ramp. Initial energy: U_g = mgh = (2.0)(9.8)(3.0) = 58.8 J.
- Step 2: Final kinetic energy: ½(2.0)(6.0)² = 36 J.
- Step 3: ΔE_thermal = 58.8 − 36 = 22.8 J ≈ 23 J.
- Step 4: Friction acts over the 5.0 m path: F_f = 22.8 J ÷ 5.0 m ≈ 4.6 N.
Answer: About 23 J of thermal energy; average friction force about 4.6 N
Common mistakes
- Counting gravity twice: as potential energy and also as external work. If Earth is in the system, use U_g; if not, use the work done by gravity.
- Assuming mechanical energy is conserved when friction or air resistance acts. Include the thermal energy term.
- Saying energy is "lost" with friction. It isn't destroyed; it becomes thermal energy and sound.
- Drawing a small bar for an energy that's zero on an energy bar chart. Leave it blank.
On the exam
- Translation Between Representations questions often ask for energy bar charts at two moments, then an equation consistent with them. Make the bars and the equation match term for term.
- Many "how fast" or "how high" questions are easiest with energy, because you don't need the path or the time. Use kinematics only if you need time.
- Expect questions that ask whether a system's energy is constant. Answer by naming the system and any external forces that do work on it.
Connected topics
Videos
Check yourself
4 questions on 3.4 Conservation of Energy. Pick an answer to see if you got it, and why.
A roller coaster car starts from rest at a height of 20 m and rolls along a frictionless track. What is its speed when it is 5.0 m above the ground? Use g = 10 m/s².
A pendulum bob is released from rest 0.45 m above its lowest point. What is its speed at the lowest point? Use g = 10 m/s² and ignore air resistance.
From the top of a building, three identical balls are thrown with the same speed: one straight up, one horizontally and one straight down. Air resistance is negligible. Which ball hits the ground with the greatest speed?
A 2.0 kg block starts from rest at the top of a ramp 5.0 m high and reaches the bottom at 8.0 m/s. How much mechanical energy was converted to thermal energy by friction? Use g = 10 m/s².
0 of 4 answered