Unit 3 · Topic 3.4

3.4 Conservation of Energy

Energy is always conserved, but whether a system's energy stays constant depends on the system you choose. If no outside forces do work and nothing like friction dissipates energy inside, mechanical energy (kinetic plus potential) stays constant. Otherwise the change equals the work done from outside, minus whatever turns into thermal energy.

Key terms

mechanical energy
conservation of energy
system
external work
energy dissipation

Mechanical energy

A system's mechanical energy is the sum of its kinetic and potential energies: $E = K + U$ . In a system where only conservative forces act internally and no outside force does work, mechanical energy stays constant:

$K_i + U_i = K_f + U_f$

Energy just shifts between forms. A pendulum trades potential energy for kinetic energy on the way down and back again on the way up.

When energy enters or leaves

The general rule is that a system's energy changes only by energy transferred across its boundary. In mechanics that transfer is work by external forces:

$\Delta K + \Delta U + \Delta E_{th} = W_{ext}$

$\Delta E_{th}$ is the increase in thermal (internal) energy. Kinetic friction between surfaces in the system turns kinetic energy into thermal energy and sound. For an object sliding a distance d, the energy dissipated is $F_{f,k}d$ . This energy isn't destroyed; it's no longer mechanical.

An isolated system (no external work) has constant total energy. Its mechanical energy stays constant only if nothing dissipates energy inside it as well.

Choosing the system

The same situation can be described two ways, and both give the same answer. For a falling ball:

System = ball + Earth: gravity is internal, so you use gravitational potential energy. $\Delta K + \Delta U_g = 0$ .
System = ball alone: there's no potential energy, and gravity is an external force doing work mgh. $\Delta K = W_{gravity}$ .
Never count gravity twice, as both a potential energy and a work. Choose one.

Solving energy problems

Energy methods shine when you care about speeds and positions but not time, and when forces vary (springs, curved tracks, gravity over large distances). The route doesn't matter for conservative forces, so a block sliding down a frictionless curved ramp lands with the same speed as one dropped from the same height.

Write the energy at the start and end, list every form present (kinetic, gravitational, spring, thermal), set up the conservation equation, then solve. In a bar chart of energy (a common representation), the bars before plus the work in must equal the bars after.

A special case: the escape speed from a planet is the launch speed that makes the total energy exactly zero, so the object just barely gets infinitely far away: $\frac{1}{2}mv^2 - \frac{GMm}{R} = 0$ , giving $v_{esc} = \sqrt{\frac{2GM}{R}}$ .

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Spring launcher up a ramp
A 0.50 kg block is pressed against a spring (k = 800 N/m), compressing it 0.10 m. When released, it slides along a frictionless track and up a ramp. How high does it rise? Use g = 9.8 m/s².
Show the solutionHide the solution
1. Step 1: System: block + spring + Earth. No friction and no external work, so mechanical energy is constant.
2. Step 2: Start: all spring energy, $\frac{1}{2}(800)(0.10)^2 = 4.0$ J. Highest point: momentarily at rest, so all gravitational energy, mgh.
3. Step 3: $h = \frac{4.0}{(0.50)(9.8)} \approx 0.82$ m.
4. Step 4: The ramp's shape doesn't matter, because only conservative forces do work.
Answer: About 0.82 m.
Example 2Calculator allowed
Sliding down a rough incline
A 2.0 kg block starts from rest and slides 5.0 m down a 30° incline with $\mu_k = 0.20$ . Find its speed at the bottom. Use g = 9.8 m/s².
Show the solutionHide the solution
1. Step 1: Height lost: $h = 5.0\sin 30^\circ = 2.5$ m, so gravitational energy drops by $(2.0)(9.8)(2.5) = 49$ J.
2. Step 2: Thermal energy from friction: $\mu_kmg\cos 30^\circ \cdot d = (0.20)(2.0)(9.8)(0.866)(5.0) \approx 17.0$ J.
3. Step 3: Energy equation: $\Delta K = 49 - 17.0 = 32.0$ J, so $v = \sqrt{\frac{2(32.0)}{2.0}} \approx 5.7$ m/s.
4. Step 4: Without friction it would be $\sqrt{2gh} = 7.0$ m/s. Friction turned about a third of the energy into thermal energy.
Answer: About 5.7 m/s.
Example 3Calculator allowed
Escape speed from Earth
Find the minimum launch speed for an object to escape Earth completely, ignoring air resistance. Earth's mass is $5.97 \times 10^{24}$ kg and its radius is $6.37 \times 10^6$ m.
Show the solutionHide the solution
1. Step 1: Just escaping means arriving at infinity (where $U_G = 0$ ) with zero speed, so the total energy is zero.
2. Step 2: $\frac{1}{2}mv^2 - \frac{GMm}{R} = 0$ , so $v = \sqrt{\frac{2GM}{R}}$ . The object's mass cancels.
3. Step 3: $v = \sqrt{\frac{2(6.67 \times 10^{-11})(5.97 \times 10^{24})}{6.37 \times 10^6}} \approx 1.12 \times 10^4$ m/s.
Answer: About 11.2 km/s.

Common mistakes

Counting gravity twice, as potential energy and as work. If Earth is in the system, use U; if not, use work.
Using conservation of mechanical energy when friction acts. Include the thermal energy, $F_{f,k}d$ .
Leaving out an energy form, like spring energy at the start or kinetic energy at the top of a projectile's path.
Using $\frac{1}{2}mv^2 = mgh$ with h measured along a slope instead of vertically.

On the exam

Free-response questions often ask you to define a system and say whether its mechanical energy is constant. Justify it by naming any external forces doing work or any friction inside.
Energy bar charts (before, work, after) are a common representation. Make the bars' sizes consistent with the conservation equation.

Connected topics

Videos

Conservation of energy (part 1) | AP Physics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
Topic 3.4 - Conservation of Energy
Lessons With LondotWatch on YouTube (opens in a new tab)
Introduction to Conservation of Mechanical Energy with Demonstrations
Flipping PhysicsWatch on YouTube (opens in a new tab)
AP Physics C - Conservation of Energy
Dan Fullerton (APlusPhysics)Watch on YouTube (opens in a new tab)
24.1 Mechanical Energy and Energy Conservation
MIT OpenCourseWareWatch on YouTube (opens in a new tab)
Energy Systems Clarified
Flipping PhysicsWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 3.4 Conservation of Energy. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A roller coaster car starts from rest at a point 20 m above the ground. Ignoring friction and air resistance, how fast is it moving when it reaches a point 5.0 m above the ground? Use g = 10 m/s².

Question 2 of 4Calculator allowed

A 0.50 kg block is pressed against a horizontal spring (k = 800 N/m), compressing it 0.10 m, and released on a frictionless floor. How fast is the block moving when it leaves the spring?

Question 3 of 4Calculator allowed

A pendulum bob on a 2.0 m string is pulled aside until the string makes 60° with the vertical, then released from rest. How fast is the bob moving at the lowest point? Use g = 10 m/s².

Question 4 of 4Calculator allowed

A 2.0 kg block is attached to the bottom of a hanging, relaxed spring (k = 400 N/m) and released from rest. How far does the block fall before it first stops? Use g = 10 m/s².

0 of 4 answered