Work, Energy, and Power Problems with Step-by-Step Answers

Overview

Here is the short version of how to approach most work energy power problems.

Core ideas to remember:

• Work: for a constant force parallel to motion, W = Fd. More generally, W = Fd cos θ.
• Kinetic energy: K = 1/2 mv².
• Gravitational potential energy: U = mgh, relative to a chosen zero level.
• Elastic potential energy: U_s = 1/2 kx².
• Work-energy theorem: W_net = ΔK.
• Mechanical energy: E = K + U.
• Power: P = W/t, and for constant force parallel to velocity, P = Fv.

Reusable checklist:

1. Write down what is known and what must be found.
2. Choose the system: one object, two objects, or object plus Earth/spring.
3. Decide whether to use work-energy or energy conservation.
4. List forces that do work: gravity, friction, spring force, applied force, tension, normal force if relevant.
5. Mark initial and final states clearly.
6. Use a sign convention and stick to it.
7. Check units before solving.
8. Ask whether the final answer is physically reasonable.

If you need a broader formula refresher while working these examples, keep a tab open to the Physics Equations Sheet by Topic: Kinematics, Forces, Energy, Waves, and Electricity.

Checklist by scenario

This section gives common mechanics scenarios, the best checklist for each one, and step-by-step answers.

1) Constant force along the direction of motion

Problem: A 5.0 kg box is pushed across a smooth floor with a constant horizontal force of 12 N for 4.0 m. Find the work done by the force and the box's change in kinetic energy.

Checklist:

• Is the force constant?
• Is it parallel to the displacement?
• Is friction absent?
• If yes, then the applied force may be the net force doing work.

Step 1: Use the work formula.
Since force and displacement are in the same direction, θ = 0°, so cos θ = 1.

W = Fd = (12)(4.0) = 48 J

Step 2: Apply the work-energy theorem.
W_net = ΔK

Because no other horizontal forces are doing work, ΔK = 48 J.

Answer: Work done = 48 J, change in kinetic energy = +48 J.

Why this method works: When net work is easy to calculate directly, the work-energy theorem is usually the fastest route.

2) Force at an angle

Problem: A student pulls a sled 6.0 m with a 20 N force at 30° above horizontal. Find the work done by the pulling force.

Checklist:

• Use the angle between force and displacement, not the angle in the diagram automatically.
• If displacement is horizontal, only the horizontal component contributes to work.

Step 1: Use W = Fd cos θ.

W = (20)(6.0) cos 30°

W ≈ 120(0.866) = 104 J

Answer: The pulling force does about 104 J of work.

Note: The vertical component of the force may change the normal force, but if the sled does not move vertically, that vertical component does no work on the sled over this displacement.

3) Friction doing negative work

Problem: A 10 kg crate slides 3.0 m across a rough floor. The friction force is 8.0 N opposite the motion. Find the work done by friction and the change in kinetic energy due to friction alone.

Checklist:

• Friction opposite motion means negative work.
• Use either a negative angle of 180° or reason from direction.

Step 1: Calculate friction work.

W_f = Fd cos 180° = (8.0)(3.0)(-1) = -24 J

Step 2: Relate to kinetic energy.

If friction is the only force doing net work, then ΔK = -24 J.

Answer: Work by friction = -24 J, so the crate's kinetic energy decreases by 24 J.

Exam habit: Negative work usually means energy is transferred out of mechanical motion into thermal energy or other forms.

4) Using gravitational potential energy

Problem: A 2.0 kg ball is dropped from rest from a height of 5.0 m. Ignore air resistance. Find its speed just before hitting the ground.

Checklist:

• If only gravity acts, conservation of mechanical energy is often easiest.
• Choose the ground as U = 0.
• From rest means initial kinetic energy is zero.

Step 1: Write energy conservation.

K_i + U_i = K_f + U_f

0 + mgh = 1/2 mv² + 0

Step 2: Cancel mass and solve.

gh = 1/2 v²

v² = 2gh = 2(9.8)(5.0) = 98

v = √98 ≈ 9.9 m/s

Answer: The ball's speed is about 9.9 m/s.

Why this is efficient: You avoid solving for acceleration and time. Energy methods often simplify vertical motion, just as a Projectile Motion Calculator Guide: Range, Time, Height, and Common Mistakes can simplify projectile checks from another angle.

5) Spring energy problem

Problem: A spring with spring constant 200 N/m is compressed 0.15 m. How much elastic potential energy is stored?

Checklist:

• Use the compression or extension from equilibrium as x.
• Square the displacement.

Step 1: Apply the spring energy formula.

U_s = 1/2 kx²

U_s = 1/2 (200)(0.15)²

U_s = 100(0.0225) = 2.25 J

Answer: Stored elastic potential energy = 2.25 J.

6) Spring launching a mass

Problem: A 0.50 kg block is launched on a frictionless surface by a spring of constant 180 N/m compressed 0.20 m. Find the block's speed after leaving the spring.

Checklist:

• Initial spring energy becomes kinetic energy if friction is negligible.
• Write the initial and final states carefully.

Step 1: Set energies equal.

1/2 kx² = 1/2 mv²

Step 2: Substitute values.

1/2 (180)(0.20)² = 1/2 (0.50)v²

90(0.04) = 0.25v²

3.6 = 0.25v²

v² = 14.4

v ≈ 3.79 m/s

Answer: The block leaves the spring at about 3.8 m/s.

7) Work-energy theorem with stopping distance

Problem: A 1200 kg car moving at 20 m/s brakes to rest. The average braking force is 6000 N opposite the motion. How far does the car travel while stopping?

Checklist:

• Stopping means final kinetic energy is zero.
• Opposite force means negative work.
• This is a classic work energy theorem problem.

Step 1: Compute the change in kinetic energy.

ΔK = K_f - K_i = 0 - 1/2 mv²

ΔK = -1/2 (1200)(20)² = -240,000 J

Step 2: Set net work equal to the change in kinetic energy.

W_net = Fd cos 180° = -6000d

-6000d = -240,000

d = 40 m

Answer: The stopping distance is 40 m.

8) Power from work and time

Problem: A machine does 900 J of work in 15 s. Find its average power.

Checklist:

• Average power uses total work divided by total time.
• Power is measured in watts, where 1 W = 1 J/s.

Step 1: Use P = W/t.

P = 900/15 = 60 W

Answer: Average power = 60 W.

9) Power from force and speed

Problem: A motor pulls an object at constant speed using a horizontal force of 50 N. The speed is 3.0 m/s. Find the power delivered.

Checklist:

• If force and velocity are parallel, use P = Fv.
• Constant speed often signals balanced forces, but the motor can still supply power to overcome resistance.

Step 1: Apply the power relation.

P = Fv = (50)(3.0) = 150 W

Answer: Power delivered = 150 W.

10) Mixed energy problem with friction

Problem: A 4.0 kg block starts from rest at the top of a ramp 2.0 m above the bottom. On the way down, friction does -10 J of work. Find the speed at the bottom.

Checklist:

• If nonconservative forces are present, use E_i + W_nc = E_f.
• Here, friction is the nonconservative force.

Step 1: Write the energy equation.

K_i + U_i + W_nc = K_f + U_f

0 + mgh - 10 = 1/2 mv² + 0

Step 2: Substitute values.

(4.0)(9.8)(2.0) - 10 = 1/2 (4.0)v²

78.4 - 10 = 2v²

68.4 = 2v²

v² = 34.2

v ≈ 5.85 m/s

Answer: The speed at the bottom is about 5.9 m/s.

What to double-check

Before you box your final answer, run through this quick audit. It catches many lost marks in physics homework help sessions and exam prep.

• Units: Mass in kilograms, distance in meters, force in newtons, energy in joules, power in watts.
• Angle choice: Use the angle between force and displacement, not force and horizontal unless those happen to match.
• Signs: Friction usually does negative work. Gravity can do positive or negative work depending on motion direction.
• Initial versus final state: Mark them clearly. Many errors come from swapping heights or speeds.
• Zero level for potential energy: Any reference level works if you stay consistent.
• Net work or single-force work: If the question asks for work by friction, do not accidentally calculate net work.
• Average versus instantaneous power: W/t gives average power over a time interval. Fv is often used for instantaneous power when force and speed are known at a moment.
• Reasonableness: A larger drop height should usually give a larger final speed if other effects are unchanged. More friction should reduce final mechanical energy.

If you tend to lose time deciding what information matters, it can help to pair energy methods with a structured triage habit like the one in Physics Problem Solving with KPI Thinking: What to Measure, What to Ignore.

Common mistakes

These are the patterns that show up again and again in work energy power problems.

Using force formulas when energy is faster

Students often start with Newton's second law, then kinematics, then time, even when a one-line energy equation would solve the problem. If the problem asks for speed, height, stopping distance, or compression, check energy first.

Forgetting that the normal force often does zero work

On a level surface, if displacement is horizontal and the normal force is vertical, the angle is 90°, so the normal force does no work. That does not mean the normal force is unimportant overall, but it may not enter the work calculation directly.

Dropping the negative sign on friction

Friction opposing motion removes mechanical energy from the system. If your calculation gives friction positive work in a standard sliding-against-friction problem, pause and recheck the direction.

Mixing up energy conservation and the work-energy theorem

They are connected but not identical. Pure mechanical energy conservation works when only conservative forces are transferring energy within the chosen system. The work-energy theorem always relates net work to change in kinetic energy. In mixed problems with friction or applied forces, many students get cleaner results by writing E_i + W_nc = E_f.

Using the wrong displacement in spring problems

Use the amount stretched or compressed from equilibrium, then square it. A small arithmetic slip matters because x² changes quickly.

Confusing speed with velocity in power relations

In many introductory problems, P = Fv assumes force and velocity point in the same direction. More generally, it is a dot product, so direction matters.

Answering with the wrong quantity

Read the prompt carefully. Some questions ask for work done by a force; some ask for net work; others ask for a change in kinetic energy, energy stored, or power output. They are related, but not interchangeable.

For students moving between mechanics and electricity topics, it helps to compare how power appears in different contexts. Mechanical power may use P = W/t or P = Fv, while electrical power uses different forms. For that contrast, see Series and Parallel Circuits Explained with Formula Sheet and Examples.

When to revisit

Use this article as a checkpoint whenever the inputs change or the stakes rise. In practice, that usually means returning to it in a few specific moments.

• Before a quiz or exam: Redo two or three examples without looking at the steps, then compare your setup.
• When a teacher changes the problem style: For example, moving from simple frictionless ramps to mixed problems with friction and springs.
• When you start a new mechanics unit: Energy methods often connect kinematics, forces, circular motion, and momentum units.
• When your class begins using calculators or digital tools differently: Recheck unit handling, stored constants, and rounding habits.
• During revision cycles: Build a one-page checklist from the patterns here and keep it with your formula sheet.

Action plan for your next study session:

1. Copy the reusable checklist into your notes.
2. Solve one work problem, one conservation of energy problem, and one power problem.
3. For each one, label the system, list forces doing work, and state why you chose your equation.
4. After solving, do the double-check audit: units, signs, angles, initial and final states.
5. If you miss a step, add that exact error to your personal mistake list.

The most effective physics study guide is not the one with the most formulas. It is the one you can return to and use quickly. If you make the checklist in this article part of your routine, work energy power problems stop feeling like separate puzzle types and start looking like variations of the same method.