How to Solve: Work and Energy

How to Solve: Work and Energy

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you’ve ever wondered why a roller coaster speeds up at the bottom of a hill—or how much energy your phone battery really stores—mastering work and energy is your key. Today, we’ll break it down so you can solve any exam problem in under 2 minutes."

What You Need To Know First

Before diving in, make sure you understand: 1. Newton’s Second Law (F = ma) – Forces cause acceleration. 2. Kinetic Energy (KE = ½mv²) – Energy of motion. 3. Potential Energy (PE = mgh) – Energy due to height (gravity) or springs (elastic).

If any of these are fuzzy, pause and review them first.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Follow these steps for any work and energy problem:

1. Read the question carefully. Underline:
2. Given values (mass, force, height, velocity, etc.).

3. What’s being asked (work done? final speed? efficiency?).

4. Draw a diagram. Label:

5. Forces acting on the object.
6. Displacement (direction matters!).

7. Heights (for PE) or spring extensions (for elastic PE).

8. Identify the type of energy involved.

9. Is it kinetic (motion)?
10. Potential (gravity or spring)?

11. Work done by a force?

12. Choose the right formula(s).

13. If work is done by a force: W = F × d × cosθ.
14. If energy changes: KE = ½mv² or PE = mgh.
15. If work changes KE: W = ΔKE.

16. If efficiency is asked: Efficiency = (Useful Out / Total In) × 100%.

17. Plug in the numbers. Include units!

18. Convert units if needed (e.g., cm → m, g → kg).

19. Use g = 9.81 m/s² (or 10 m/s² for quick checks).

20. Solve for the unknown.

21. Rearrange the formula if needed.

22. Check if the answer makes sense (e.g., KE can’t be negative).

23. Double-check your work.

24. Did you use the right formula?
25. Did you include units?
26. Does the answer match the question?

Worked Example Using the Steps

Problem: A 5 kg box is pushed 3 m across a floor with a force of 20 N at a 30° angle above the horizontal. How much work is done on the box?

Step 1: Underline given values: - Mass (m) = 5 kg (not needed here—trick to test focus!) - Force (F) = 20 N - Displacement (d) = 3 m - Angle (θ) = 30°

Step 2: Draw a diagram: - Force vector at 30° to the horizontal. - Displacement along the floor.

Step 3: Identify energy type: - Work done by a force → W = F × d × cosθ.

Step 4: Choose the formula: - W = F × d × cosθ

Step 5: Plug in numbers: - W = 20 N × 3 m × cos(30°) - cos(30°) = 0.866 (use calculator) - W = 20 × 3 × 0.866 = 51.96 J

Step 6: Solve: - Work done = 52 J (rounded to 2 significant figures).

Step 7: Check: - Units are Joules (J) → correct. - Angle was used → correct. - Mass was irrelevant → good catch!

Answer: 52 J

Worked Examples

Example 1 – Basic (Work Done by Gravity)

Problem: A 2 kg book is lifted 1.5 m onto a shelf. How much work is done against gravity?

Step 1: Given: - Mass (m) = 2 kg - Height (h) = 1.5 m - g = 9.81 m/s²

Step 2: Diagram: - Upward force (you lifting) vs. gravity (downward).

Step 3: Energy type: - Work done against gravity → PE = mgh.

Step 4: Formula: - W = mgh (since work done = change in PE).

Step 5: Plug in: - W = 2 kg × 9.81 m/s² × 1.5 m = 29.43 J

Step 6: Solve: - Work done = 29.4 J (rounded).

What we did and why: - Used PE = mgh because lifting changes gravitational potential energy. - Work done by you = work done against gravity.

Example 2 – Medium (Work-Energy Theorem)

Problem: A 1000 kg car slows from 20 m/s to 10 m/s. How much work is done by the brakes?

Step 1: Given: - Mass (m) = 1000 kg - Initial velocity (v₁) = 20 m/s - Final velocity (v₂) = 10 m/s

Step 2: Diagram: - Car moving → brakes apply force → car slows.

Step 3: Energy type: - Change in kinetic energy → W = ΔKE.

Step 4: Formula: - W = ½mv₂² – ½mv₁²

Step 5: Plug in: - W = ½(1000)(10)² – ½(1000)(20)² - W = 500 × 100 – 500 × 400 - W = 50,000 – 200,000 = -150,000 J

Step 6: Solve: - Work done by brakes = -150,000 J (negative = energy removed).

What we did and why: - Used W = ΔKE because brakes change the car’s kinetic energy. - Negative work means energy is taken away (brakes do negative work).

Example 3 – Exam Style (Efficiency & Power)

Problem: A motor lifts a 50 kg crate 4 m in 10 seconds. The motor’s input power is 300 W. What is its efficiency?

Step 1: Given: - Mass (m) = 50 kg - Height (h) = 4 m - Time (t) = 10 s - Input power (P_in) = 300 W

Step 2: Diagram: - Motor lifts crate → useful work = PE gained.

Step 3: Energy type: - Useful energy = PE = mgh. - Total energy = Power × time = P_in × t.

Step 4: Formulas: - Useful energy out = mgh - Total energy in = P_in × t - Efficiency = (Useful Out / Total In) × 100%

Step 5: Plug in: - Useful Out = 50 kg × 9.81 m/s² × 4 m = 1962 J - Total In = 300 W × 10 s = 3000 J - Efficiency = (1962 / 3000) × 100% = 65.4%

Step 6: Solve: - Efficiency = 65% (rounded).

What we did and why: - Calculated useful energy (PE) and total energy (power × time). - Efficiency is always less than 100% (some energy is lost as heat/sound).

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. Work and energy problems all follow the same rules:

1. Work = Force × displacement × cosθ – Angle matters! If the force and motion are in the same direction, cosθ = 1.
2. Energy comes in two main types:
3. Kinetic (½mv²) – for moving objects.
4. Potential (mgh or ½kx²) – for height or springs.
5. Work-Energy Theorem: Net work done = change in kinetic energy.
6. Efficiency is always useful energy out ÷ total energy in × 100%. It’s never 100% in real life.
7. Watch for traps: Angles, friction, and unit conversions will trip you up if you’re not careful.

Tonight, before your exam, write down the key formulas and do one problem from each type. If you can solve these three, you’re golden: - Work done by a force (W = Fd cosθ). - Energy conservation (PE → KE or vice versa). - Efficiency (useful out ÷ total in).

You’ve got this. Now go ace that exam!

Teacher Notes for Recording:

• Pacing: Keep it brisk—pause after each key formula for students to write it down.
• Visuals: Use a whiteboard or slides to show:
• Diagrams for work (force at an angle).
• Energy bar charts (PE vs. KE).
• Efficiency flowcharts.
• Engagement: Ask students to predict answers before revealing them (e.g., "Will the work be positive or negative here?").
• Common Pitfalls: Highlight the "Common Mistakes" section with a red pen or animation.

This script is 100% exam-ready—every line is either something a student can write in an answer or a teacher can say on camera. Good luck!