Energy, Work, and Power
Key Takeaways
- Work = Force × Distance (in the direction of force). No movement = no work, regardless of effort.
- Kinetic energy is energy of motion: KE = 1/2 mv². Doubling speed quadruples kinetic energy.
- Potential energy is stored energy: gravitational PE = mgh, elastic PE = 1/2 kx².
- Conservation of energy: total energy in a closed system remains constant — it only changes form.
- Power = Work / Time. Power measures how quickly work is done, not how much work is done.
These concepts are interconnected and appear frequently on the MCT. Understanding the relationships between them is more important than memorizing individual formulas.
Work
Definition
Work = Force × Distance × cos(θ)
Where θ is the angle between the force direction and the movement direction.
For most OAR questions, the force and movement are in the same direction (θ = 0°, cos(0°) = 1):
W = F × d
| Variable | Unit |
|---|---|
| Work (W) | Joules (J) or foot-pounds (ft·lb) |
| Force (F) | Newtons (N) or pounds (lb) |
| Distance (d) | Meters (m) or feet (ft) |
When Is Work Done?
| Scenario | Work Done? | Why |
|---|---|---|
| Push a box 5 meters across a floor | Yes | Force in the direction of movement |
| Hold a 50 kg weight overhead, standing still | No | No distance moved |
| Carry a box horizontally at constant speed | No work against gravity | Force (up) is perpendicular to movement (forward) |
| Lift a box 2 meters off the ground | Yes | Force (up) in direction of movement (up) |
Worked Examples
Example 1: How much work is done pushing a crate 10 meters with a force of 200 N? W = 200 × 10 = 2,000 J
Example 2: A sailor lifts a 30 kg supply box 1.5 meters. How much work is done?
- Force = weight = 30 × 10 = 300 N
- W = 300 × 1.5 = 450 J
Kinetic Energy
KE = 1/2 × m × v²
Kinetic energy is the energy an object has because it is moving.
| Variable | Unit |
|---|---|
| KE | Joules (J) |
| m (mass) | kg |
| v (velocity) | m/s |
The v² Relationship
Because velocity is squared, speed has an outsized effect on kinetic energy:
| Speed | KE Multiplier |
|---|---|
| 1× | 1× |
| 2× | 4× |
| 3× | 9× |
| 4× | 16× |
This is why car accidents at high speed are so much more dangerous. Doubling your speed quadruples the energy of impact.
Example: A 1,500 kg vehicle travels at 20 m/s. What is its kinetic energy? KE = 1/2 × 1,500 × 20² = 1/2 × 1,500 × 400 = 300,000 J = 300 kJ
Potential Energy
Gravitational Potential Energy
PE = m × g × h
Energy stored by an object's position above a reference point.
| Variable | Unit |
|---|---|
| PE | Joules (J) |
| m | kg |
| g | 9.8 m/s² (≈ 10 m/s²) |
| h | meters (height above reference) |
Example: A 20 kg box sits on a shelf 3 meters high. PE = 20 × 10 × 3 = 600 J
Elastic Potential Energy
PE = 1/2 × k × x²
Energy stored in a compressed or stretched spring.
| Variable | Unit |
|---|---|
| k | Spring constant (N/m) |
| x | Displacement from natural length (m) |
Conservation of Energy
Total energy in a closed system remains constant.
Energy is never created or destroyed — it only changes form.
Common Energy Conversions
| From | To | Example |
|---|---|---|
| Potential → Kinetic | A ball falling | PE at top converts to KE at bottom |
| Kinetic → Potential | A ball thrown upward | KE at launch converts to PE at peak |
| Chemical → Kinetic | Fuel in an engine | Chemical energy becomes motion |
| Kinetic → Heat | Braking a car | Motion energy becomes thermal energy via friction |
| Electrical → Light + Heat | A light bulb | Electrical energy converts to light and heat |
Conservation Problems
Example: A 2 kg ball is dropped from 5 meters. What is its speed just before hitting the ground? (Ignore air resistance.)
At the top: PE = mgh = 2 × 10 × 5 = 100 J, KE = 0 At the bottom: PE = 0, KE = 100 J
KE = 1/2 mv² → 100 = 1/2 × 2 × v² → v² = 100 → v = 10 m/s
Power
Power = Work / Time
Power measures the rate at which work is done.
| Variable | Unit |
|---|---|
| Power (P) | Watts (W) = J/s |
| Work (W) | Joules (J) |
| Time (t) | Seconds (s) |
Also: P = F × v (force times velocity, for constant speed)
Comparing Power
| Machine | Work Done | Time | Power |
|---|---|---|---|
| Machine A | 1,000 J | 10 s | 100 W |
| Machine B | 1,000 J | 5 s | 200 W |
| Machine C | 2,000 J | 10 s | 200 W |
Machine A and Machine B do the same work, but Machine B does it faster (more power). Machine B and Machine C have the same power, but Machine C does more total work.
Example: A crane lifts a 500 kg beam 20 meters in 25 seconds. What power does it deliver?
- Work = mgh = 500 × 10 × 20 = 100,000 J
- Power = 100,000 / 25 = 4,000 W = 4 kW
Horsepower
1 horsepower (hp) ≈ 746 watts
This conversion occasionally appears on mechanical comprehension tests.
Efficiency
Efficiency = (Useful output / Total input) × 100%
No real machine is 100% efficient — some energy is always lost to friction, heat, or sound.
Example: A motor uses 500 J of electrical energy to do 400 J of useful work. Efficiency = (400/500) × 100% = 80%
A soldier pushes a 50 kg crate 8 meters across a floor with a force of 150 N. How much work is done?
If you double the speed of a moving object, its kinetic energy:
A 10 kg object is dropped from 20 meters. What is its speed just before impact? (Use g = 10 m/s², ignore air resistance.)
Two machines both lift a 100 kg weight 5 meters. Machine A takes 10 seconds; Machine B takes 20 seconds. Which statement is correct?
A motor uses 800 J of energy to perform 600 J of useful work. What is its efficiency?