# Gravitational and Elastic Potential Energy

> AP Physics 1 · AP Physics 1 CED Unit 4: Energy
> Source: https://www.owlsprep.com/study/ap-physics-1-u4-gravitational-and-elastic-potential-energy/

This aligned AP Physics 1 guide covers gravitational potential energy near Earth's surface, elastic potential energy for ideal springs, and application of conservation of mechanical energy to systems with both potential energy types.

**Prerequisites:** Work and the work-energy theorem for kinetic energy; Hooke's Law for ideal springs; Definition of a closed system for energy analysis

## Learning objectives

- Define gravitational and elastic potential energy as system properties
- Calculate changes in gravitational potential energy near Earth's surface
- Calculate elastic potential energy for ideal springs
- Apply conservation of mechanical energy to problems involving both potential energy types

## Core Concepts of Potential Energy

Potential energy is stored energy in a closed system arising from the relative positions of interacting objects. Gravitational potential energy is stored due to gravitational interaction between masses; for AP Physics 1, we only work with values near Earth's surface where gravity is approximately constant. Elastic potential energy is stored when an elastic object like an ideal spring is stretched or compressed from equilibrium. By convention, we use $U$ for potential energy, with $U_g$ for gravitational and $U_s$ (or $U_{el}$) for elastic. Only changes in potential energy are physically meaningful, so we can choose any reference point to set $U=0$. This topic makes up ~6-9% of your total AP Physics 1 exam score, appearing in both MCQ and FRQ.

**Potential Energy** — Stored energy in a closed system that arises from the relative positions of interacting objects

*Example:* Gravitational potential energy from the height of an object above Earth, elastic potential energy from stretching a spring

## Gravitational Potential Energy Near Earth's Surface

Gravitational potential energy describes stored energy from separation between masses in a gravitational field. Near Earth's surface, the gravitational force on a mass $m$ is approximately constant, pointing downward. The change in gravitational potential energy of the mass-Earth system equals the negative of the work done by gravity when the mass changes position:

$$\Delta U_g = -W_g$$

If the mass changes height by $\Delta y = y_{final} - y_{initial}$ (with upward as positive), gravity does work $W_g = -mg\Delta y$. Substituting gives:

$$\Delta U_g = mg\Delta y$$

If we choose a reference point where $U_g = 0$, we can write $U_g = mgy$, where $y$ is the height of the mass relative to that reference. Increasing the height of the mass increases stored gravitational potential energy, which makes intuitive sense, as the stored energy can be converted to motion when the object falls.

**Worked example:** A 62 kg hiker climbs from the base of a trail at 1200 m elevation to the summit at 2450 m elevation. What is the change in the gravitational potential energy of the hiker-Earth system? If the hiker sets $U_g = 0$ at the base, what is $U_g$ at the summit?

1. Identify known values: $m = 62\ \text{kg}$, $g = 9.8\ \text{m/s}^2$, $\Delta y = 2450\ \text{m} - 1200\ \text{m} = 1250\ \text{m}$.
2. Use the gravitational potential energy change formula:
3. $$\Delta U_g = mg\Delta y$$
4. Calculate the change:
5. $$\Delta U_g = (62)(9.8)(1250) = 759500\ \text{J} \approx 7.6 \times 10^5\ \text{J}$$
6. If $U_{g,\text{base}} = 0$, then $U_{g,\text{summit}} = U_{g,\text{base}} + \Delta U_g = 0 + 7.6 \times 10^5\ \text{J} = 7.6 \times 10^5\ \text{J}$.

> **Exam tip:** Always confirm that $\Delta y$ is the change in vertical height, not horizontal distance or trail length—AP 1 questions often give trail length to test this distinction.

## Elastic Potential Energy for Ideal Springs

Elastic potential energy is stored in an ideal spring (or other elastic material) when it is stretched or compressed from its equilibrium (relaxed) position. An ideal spring obeys Hooke's Law:

$$F_s = -kx$$

where $x$ is displacement from equilibrium, and $k$ is the spring constant (a measure of stiffness, units N/m). To find elastic potential energy, we use the relation that change in potential energy equals negative work done by the conservative spring force. The work done by the spring moving from equilibrium ($x=0$) to displacement $x$ is $W_s = -\frac{1}{2}kx^2$, so we get the standard formula, with $U_s = 0$ at equilibrium (the standard AP 1 convention):

$$U_s = \frac{1}{2}kx^2$$

Notice that $x$ is squared, so stretching a spring by 0.1 m stores the same amount of energy as compressing it by 0.1 m. Stiffer springs (higher $k$) store more energy for the same displacement, and energy grows with the square of displacement, so doubling displacement quadruples stored energy.

**Worked example:** A 0.25 kg block is pressed against a horizontal spring with spring constant 120 N/m, compressing it 0.15 m from equilibrium. How much elastic potential energy is stored in the block-spring system? If the spring is compressed twice as far, by what factor does stored energy increase?

1. Identify known values: $k = 120\ \text{N/m}$, $x = 0.15\ \text{m}$, $U_s = 0$ at $x=0$.
2. Use the elastic potential energy formula:
3. $$U_s = \frac{1}{2}kx^2 = 0.5(120)(0.15)^2 = 1.35\ \text{J} \approx 1.4\ \text{J}$$
4. For twice the compression, new displacement $x' = 2x$, so:
5. $$U_s' = \frac{1}{2}k(2x)^2 = 4\left(\frac{1}{2}kx^2\right) = 4U_s$$
6. Stored energy increases by a factor of 4.

> **Exam tip:** Never use the total length of the spring for $x$—$x$ is always displacement from equilibrium (relaxed length), so you must subtract the relaxed length from the stretched/compressed length to get $x$.

## Conservation of Mechanical Energy with Multiple Potential Energies

Total mechanical energy of a closed system is the sum of kinetic energy, gravitational potential energy, and elastic potential energy:

$$E_{mech} = K + U_g + U_s$$

If no non-conservative forces (friction, air resistance, external pushes) do net work on the system, total mechanical energy is conserved. This means initial total mechanical energy equals final total mechanical energy:

$$K_i + U_{g,i} + U_{s,i} = K_f + U_{g,f} + U_{s,f}$$

This is one of the most useful problem-solving tools in AP Physics 1, because it lets you find speed or position without calculating acceleration or forces at every point along the motion. To simplify calculations, always choose a convenient zero reference for $U_g$, usually setting $U_g = 0$ at the lowest point of motion, so that term becomes zero in the final calculation.

**Worked example:** A 0.50 kg ball is dropped from rest from a height of 2.0 m above a vertical relaxed spring. The spring compresses 0.25 m before the ball momentarily stops. What is the spring constant of the spring, assuming no air resistance?

1. Define the system as the ball, Earth, and spring (no non-conservative work, so energy is conserved). Set $U_g = 0$ at the maximum compression point (where the ball stops), so $U_{g,f} = 0$.
2. Identify initial and final energies: Initial state: $K_i = 0$ (dropped from rest), $U_{s,i} = 0$ (spring is relaxed), $U_{g,i} = mg(2.0\ \text{m} + 0.25\ \text{m}) = mg(2.25\ \text{m})$ (total height change from initial to final includes compression). Final state: $K_f = 0$ (momentarily stopped), $U_{s,f} = \frac{1}{2}k(0.25\ \text{m})^2$, $U_{g,f} = 0$.
3. Substitute into conservation of energy:
4. $$0 + mg(2.25) + 0 = 0 + 0 + \frac{1}{2}k(0.25)^2$$
5. Solve for $k$:
6. $$k = \frac{2mg(2.25)}{(0.25)^2} = \frac{2(0.50)(9.8)(2.25)}{0.0625} \approx 350\ \text{N/m}$$

**Check your understanding**

Test your understanding of elastic potential energy ratios:

1. A student stretches two springs. Spring 1 has a spring constant $2k$ and is stretched by distance $x$. Spring 2 has a spring constant $k$ and is stretched by distance $3x$. How does the elastic potential energy stored in Spring 1 compare to Spring 2?

   - A) $U_1 = \frac{2}{9} U_2$
   - B) $U_1 = \frac{1}{3} U_2$
   - C) $U_1 = \frac{2}{3} U_2$
   - D) $U_1 = 2 U_2$

   *Why:* Correct. Using $U_s = \frac{1}{2}kx^2$, $U_1 = kx^2$, $U_2 = \frac{9}{2}kx^2$, so $\frac{U_1}{U_2} = \frac{2}{9}$.

**Worked example:** A 0.40 kg toy car moves along a frictionless track that starts with a hill, then has a horizontal section ending in a spring bumper. The car starts from rest at the top of the hill, 1.5 m above the horizontal section. The spring bumper has a spring constant of 80 N/m. (a) Calculate the speed of the car when it reaches the horizontal section, before it hits the spring. (b) Calculate the maximum compression of the spring when the car momentarily stops. (c) A student claims that if the car started from a height twice as high, the maximum compression of the spring would also double. Do you agree with this claim? Justify your answer.

1. Part (a): Set $U_g = 0$ at the horizontal section. By energy conservation, all initial gravitational potential energy converts to kinetic energy:
2. $$mgh = \frac{1}{2}mv^2$$
3. Mass cancels out, so solve for $v$:
4. $$v = \sqrt{2gh} = \sqrt{2(9.8)(1.5)} \approx 5.4\ \text{m/s}$$
5. Part (b): At maximum compression, all gravitational potential energy is converted to elastic potential energy:
6. $$mgh = \frac{1}{2}kx^2$$
7. Solve for $x$:
8. $$x = \sqrt{\frac{2mgh}{k}} = \sqrt{\frac{2(0.40)(9.8)(1.5)}{80}} = \sqrt{0.147} \approx 0.38\ \text{m}$$
9. Part (c): The claim is incorrect. From the relation above, $x = \sqrt{\frac{2mgh}{k}}$, so $x$ is proportional to $\sqrt{h}$, not $h$. If $h$ doubles, $x' = \sqrt{2}x \approx 1.41x$, so compression increases by a factor of $\sqrt{2}$, not 2.

> **Exam tip:** Always sketch initial and final positions to confirm all displacement/height changes—students almost always forget to add the spring compression to the total height change in this type of problem.

## Common pitfalls

- **Wrong:** Using the total length of a spring instead of displacement from equilibrium in the elastic potential energy formula.
  - Why it fails: Questions often give both relaxed and stretched length to test understanding, and students default to the larger number given.
  - Correct: Calculate $x$ as $|\text{final length} - \text{relaxed equilibrium length}|$ every time before plugging into $U_s = \frac{1}{2}kx^2$.
- **Wrong:** Forgetting to add the spring compression/extension to the total height change when solving energy problems with a falling object on a vertical spring.
  - Why it fails: Students stop at the height of the object above the relaxed spring, and ignore the additional distance the object falls while compressing the spring.
  - Correct: Always measure the total change in height from the initial position of the object to its final position, regardless of what the spring is doing.
- **Wrong:** Dropping the negative sign for $U_g$ when the object is below the chosen zero reference point.
  - Why it fails: Students assume potential energy can never be negative, forgetting only changes in potential energy are physically meaningful.
  - Correct: Keep the sign of $U_g$ equal to the sign of $y$ relative to your zero reference; the change in potential energy will still be correct.
- **Wrong:** Claiming stretching a spring twice as far stores twice as much elastic potential energy.
  - Why it fails: Students confuse the linear force-displacement Hooke's Law relation with the quadratic energy-displacement relation.
  - Correct: Remember that energy depends on the square of displacement, so doubling displacement quadruples stored energy.
- **Wrong:** Treating potential energy as a property of a single object instead of a system of interacting objects.
  - Why it fails: Textbooks often use shorthand like "the hiker's potential energy," leading to this mistake, which AP 1 explicitly tests.
  - Correct: Explicitly reference the system: gravitational potential energy belongs to the hiker-Earth system, elastic potential energy belongs to the spring-mass system.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Change in Gravitational Potential Energy | $\Delta U_g = mg\Delta y$ | Only applies near Earth's surface where $g$ is constant. $\Delta y = y_{final} - y_{initial}$, upward = positive. Only $\Delta U$ is physically meaningful. |
| Gravitational Potential Energy (with reference) | $U_g = mgy$ | $y$ = height relative to chosen $U_g=0$ reference. Can be negative for positions below the reference. |
| Elastic Potential Energy (Ideal Spring) | $U_s = \frac{1}{2}kx^2$ | $x$ = displacement from equilibrium. $U_s=0$ at equilibrium, same energy for compression or stretch. |
| Total Mechanical Energy | $E_{mech} = K + U_g + U_s$ | $K = \frac{1}{2}mv^2$ = total kinetic energy of all moving objects in the system. |
| Conservation of Mechanical Energy | $K_i + U_{g,i} + U_{s,i} = K_f + U_{g,f} + U_{s,f}$ | Only applies if no net work is done by non-conservative forces (friction, external pushes). |
| Potential Energy-Work Relation | $\Delta U = -W_{conservative}$ | Change in potential energy equals negative work done by the conservative force (gravity, spring force). |
| Potential Energy System Property | $U_g$ belongs to system of interacting masses | Never attribute $U_g$ to a single mass; AP 1 tests this conceptual point regularly. |

## What's next

This topic is the foundation for all energy-based problem solving in the rest of AP Physics 1. Next, you will extend energy conservation to account for work done by non-conservative forces like friction, which converts mechanical energy to thermal energy, leading to the full, general work-energy theorem. Without mastering how to correctly calculate gravitational and elastic potential energy, you cannot solve energy problems for collisions, simple harmonic motion, or rotational systems that come later in the course. This topic also builds the core understanding of energy conservation as a fundamental law that applies to all physical systems in AP Physics 1 and beyond, making it a high-priority topic for exam preparation.

- [Conservation of Mechanical Energy](https://www.owlsprep.com/study/ap-physics-1-u4-conservation-of-mechanical-energy/)
- [Power](https://www.owlsprep.com/study/ap-physics-1-u4-power/)
- [Momentum Overview](https://www.owlsprep.com/study/ap-physics-1-u5-overview/)

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