# AP Physics 1 Work and Kinetic Energy

> AP Physics 1 · Unit 4: Energy
> Source: https://www.owlsprep.com/study/ap-physics-1-u4-work-and-kinetic-energy/

This module covers mechanical work, kinetic energy, the work-kinetic energy theorem, work from variable forces on F-x graphs, and power, aligned to AP Physics 1 CED Unit 4, with worked examples and exam tips.

**Prerequisites:** Resolving vectors into parallel and perpendicular components; Newton's second law and constant acceleration kinematics; Calculating area of simple geometric shapes

## Learning objectives

- Define mechanical work and kinetic energy
- Calculate work done by constant and variable forces
- Apply the work-kinetic energy theorem to solve motion problems
- Calculate average and instantaneous power
- Solve AP-style work and energy problems

## Mechanical Work by a Constant Force

**Mechanical Work** — Scalar measure of energy transferred by a force acting over a displacement, equal to the dot product of force and displacement vectors.

*Notation:* W

$$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$$

Where $\theta$ is the angle between the force and displacement vectors, and the SI unit of work is the joule ($1\ \text{J} = 1\ \text{N} \cdot \text{m}$). Only the component of force parallel to displacement does work. Perpendicular components do zero work because $\cos 90^\circ = 0$. Work is positive when energy is added to the object ($\theta < 90^\circ$) and negative when energy is removed ($\theta > 90^\circ$). Net work is the algebraic sum of work done by all individual forces.

**Worked example:** A student pulls a 12 kg sled across frictionless horizontal ice with a rope that makes a 30° angle with the horizontal. Tension in the rope is 40 N, and the sled moves 5.0 m horizontally. What is the work done on the sled by tension?

1. Identify given values, note gravity and normal force are perpendicular to displacement so they do zero work, only tension contributes.

   $$F_T = 40\ \text{N}, \theta = 30^\circ, d = 5.0\ \text{m}$$
2. Apply the constant-force work formula:
3. Substitute values and calculate:

   $$W = (40)(5.0)\cos(30^\circ) = 200(0.866) = 173.2\ \text{J}$$
4. Round to two significant figures consistent with given values:

   $$W = 170\ \text{J}$$

> **Exam tip:** Always confirm which axis the problem's given angle is measured from. If the angle is given from the vertical instead of the horizontal, use $\sin\theta$ instead of $\cos\theta$ to get the parallel component of force.

## Kinetic Energy and the Work-Energy Theorem

**Kinetic Energy** — Energy an object possesses due to its motion, always non-negative and a scalar quantity.

*Notation:* K

$$K = \frac{1}{2}mv^2$$

**Work-Energy Theorem** — The net work done on an object by all forces equals the change in the object's kinetic energy, valid for both constant and variable forces.

*Notation:* W_{\text{net}} = \Delta K

$$W_{\text{net}} = \Delta K = K_f - K_i = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$

This theorem is a powerful alternative to Newtonian kinematics for problems relating speed and displacement, eliminating the need to calculate acceleration first.

**Worked example:** Using the sled from the previous example, if the sled starts from rest, what is its speed after moving 5.0 m?

1. Net work equals work done by tension, since friction is zero and other forces do zero work. Initial kinetic energy is zero.

   $$W_{\text{net}} = 173\ \text{J}, K_i = 0$$
2. Apply the work-energy theorem:

   $$W_{\text{net}} = \frac{1}{2}mv_f^2 - 0$$
3. Rearrange to solve for final speed and substitute values:

   $$v_f = \sqrt{\frac{2W_{\text{net}}}{m}} = \sqrt{\frac{2(173)}{12}} = \sqrt{28.8} \approx 5.4\ \text{m/s}$$

> **Exam tip:** If a problem gives displacement and asks for speed, always check if the work-energy theorem is faster than kinematics. It will save you 1-2 minutes on most MCQ questions.

## Work Done by a Variable Force

When force varies with position (e.g., spring force, changing applied force), the constant-force work formula does not apply. For AP Physics 1, work done by a variable force is equal to the total area under a force vs. position ($F$ vs. $x$) graph between the initial and final position.

> **info**
>
> Area above the x-axis = positive work; Area below the x-axis = negative work. Split the graph into simple geometric shapes (triangles, rectangles, trapezoids) to calculate total area.

**Worked example:** A variable force acts on a 2 kg cart moving along the x-axis, with the force profile: $F$ increases linearly from 0 N at $x=0$ to 8 N at $x=2$ m, stays constant at 8 N from $x=2$ m to $x=5$ m, then decreases linearly back to 0 N at $x=7$ m. What is the total work done by the force between $x=0$ and $x=7$ m?

1. Split the graph into three regions to calculate area: 0-2 m, 2-5 m, 5-7 m.
2. Region 1 (0-2 m, triangle):

   $$W_1 = \frac{1}{2} \times 2\ \text{m} \times 8\ \text{N} = 8\ \text{J}$$
3. Region 2 (2-5 m, rectangle):

   $$W_2 = (5-2)\ \text{m} \times 8\ \text{N} = 24\ \text{J}$$
4. Region 3 (5-7 m, triangle):

   $$W_3 = \frac{1}{2} \times 2\ \text{m} \times 8\ \text{N} = 8\ \text{J}$$
5. Sum the areas to get total work:

   $$W_{\text{total}} = 8 + 24 + 8 = 40\ \text{J}$$

> **Exam tip:** If the force crosses from positive to negative on the $F-x$ graph, don't forget to subtract the area of the negative region, don't just add all areas regardless of sign.

## Power

**Power** — The rate at which work is done (or energy is transferred) between systems. SI unit is the watt ($1\ \text{W} = 1\ \text{J/s}$).

*Notation:* P

Average power over a time interval $\Delta t$ is given by:

$$P_{\text{avg}} = \frac{W}{\Delta t}$$

For instantaneous power (power at a specific moment), when force $F$ is parallel to velocity $v$, the formula simplifies to:

$$P = Fv$$

This is commonly used for problems involving engines, vehicles, or human movement where power output is given.

**Worked example:** A 1500 kg car accelerates from rest to 20 m/s, with negligible friction. If the car's engine delivers an average power of 40 kW, how much time does the acceleration take?

1. The net work done by the engine equals the change in kinetic energy of the car:

   $$W = \Delta K = \frac{1}{2}mv_f^2 - 0 = 0.5(1500)(20)^2 = 300{,}000\ \text{J} = 300\ \text{kJ}$$
2. Convert average power to standard SI units (watts):

   $$40\ \text{kW} = 40{,}000\ \text{W}$$
3. Rearrange the average power formula to solve for time:

   $$\Delta t = \frac{W}{P_{\text{avg}}} = \frac{300000}{40000} = 7.5\ \text{s}$$

> **Exam tip:** Always convert kilowatts to watts (multiply by 1000) before calculating energy or time. A common mistake leaves power in kilowatts and gets a time 1000 times smaller than the correct value.

## Common pitfalls

- **Wrong:** Using the full magnitude of an angled force in the work formula, omitting the $\cos\theta$ term.
  - Why it fails: Students memorize the simplified $W=Fd$ for parallel forces and forget to adjust for angled forces.
  - Correct: Always resolve the force into parallel and perpendicular components, and only use the parallel component for work calculations.
- **Wrong:** Plugging work done by a single force into the work-energy theorem instead of net work.
  - Why it fails: Students confuse "work done by the applied force" with "net work from all forces".
  - Correct: Always sum work done by every force (including friction, gravity, and normal force) to get $W_{\text{net}}$ before applying the theorem.
- **Wrong:** Treating kinetic energy as a vector and adding it via vector components.
  - Why it fails: Students are used to working with velocity and force vectors, so they carry over vector addition to kinetic energy.
  - Correct: Remember kinetic energy is a scalar; add magnitudes directly, no components required.
- **Wrong:** Counting area below the x-axis on a $F-x$ graph as positive work.
  - Why it fails: Students remember "area equals work" but forget force direction changes the sign.
  - Correct: Always assign a negative sign to area below the x-axis when calculating total work.
- **Wrong:** Using $P=Fv$ to calculate average power for an accelerating object.
  - Why it fails: Students memorize the simplified power formula and use it for any problem.
  - Correct: Only use $P=Fv$ for instantaneous power; use $P_{\text{avg}} = W/\Delta t$ for average power over a time interval.

## Cheatsheet

| Category | Formula | Notes |
| --- | --- | --- |
| Work by constant force | $W = Fd\cos\theta$ | $\theta$ = angle between force and displacement; only parallel force does work |
| Net Work | $W_{\text{net}} = \sum W_i$ | Sum of work done by all forces acting on the object |
| Kinetic Energy | $K = \frac{1}{2}mv^2$ | Scalar, always non-negative; units are joules (J) |
| Work-Energy Theorem | $W_{\text{net}} = \Delta K = K_f - K_i$ | Applies to constant and variable forces |
| Work by variable force | $W = \text{Area under } F-x \text{ graph}$ | Area above x-axis = positive work; area below = negative work |
| Average Power | $P_{\text{avg}} = \frac{W}{\Delta t}$ | Units are watts (W) = 1 J/s |
| Instantaneous Power | $P = Fv$ | Only valid when force is parallel to velocity |

## What's next

Work and kinetic energy is the fundamental prerequisite for all remaining energy topics in AP Physics 1 Unit 4. Next, you will extend the work-energy theorem to include potential energy, stored energy due to position, leading to the principle of conservation of energy for closed systems. Without mastering how to calculate work and apply the work-energy theorem, you will not be able to correctly solve problems involving gravitational or elastic potential energy, which make up the majority of energy-related FRQ questions on the AP exam. This topic also connects directly to power in electric circuits later in the course, where power is defined identically as the rate of energy transfer.

- [AP Physics 1 Work-Energy Theorem](https://www.owlsprep.com/study/ap-physics-1-u4-work-energy-theorem/)
- [Gravitational and Elastic Potential Energy](https://www.owlsprep.com/study/ap-physics-1-u4-gravitational-and-elastic-potential-energy/)
- [Conservation of Mechanical Energy](https://www.owlsprep.com/study/ap-physics-1-u4-conservation-of-mechanical-energy/)

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