Playground
Energy bar diagram showing initial KE, work done, and final KE with color-coded transfers.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Net Workoutput Total work done by all forces on the object | J | ML²T⁻² | -10000 – 10000 | |
| Mass Mass of the object | kg | M | 0.1 – 2000 | |
| Final Velocity Velocity of the object after work is done | m/s | LT⁻¹ | 0 – 100 | |
| Initial Velocity Velocity of the object before work is done | m/s | LT⁻¹ | 0 – 100 |
Deep dive
Derivation
Start with F_net = ma. Multiply both sides by ds: F_net·ds = ma·ds = m(dv/dt)·ds = mv·dv. Integrate from v_i to v_f: W_net = ∫mv·dv = ½mv_f² - ½mv_i².
Experimental verification
Air track experiments with photogates, crash test energy absorption measurements.
Common misconceptions
- Work done by individual forces can be positive, negative, or zero — only the net matters
- Normal force on a flat surface does zero work (perpendicular to motion)
- Friction always does negative work on the sliding object
Real-world applications
- Vehicle braking distance estimation
- Roller coaster energy budgets
- Impact engineering
- Sports performance analysis
Worked examples
Car braking distance
Given:
- m:
- 1500
- v_i:
- 30
- v_f:
- 0
- F_brake:
- 10000
Find: stopping distance d
Solution
W = ΔKE → F·d = ½mv² → d = mv²/(2F) = 1500×900/(2×10000) = 67.5 m
Accelerating a sprinter
Given:
- m:
- 75
- v_i:
- 0
- v_f:
- 10
Find: W_net
Solution
W = ½(75)(10²) - 0 = 3750 J
Scenarios
What if…
- scenario:
- What if braking force doubles?
- answer:
- Stopping distance halves. With 20,000 N of braking force: d = 675000/20000 = 33.75 m.
- scenario:
- What if the road is inclined upward?
- answer:
- Gravity does negative work too, assisting braking. W_gravity = -mgh. The car stops in a shorter distance.
- scenario:
- What about friction doing work?
- answer:
- Friction converts KE to thermal energy. W_friction = -μmgd. This is why brakes heat up during hard stops.
Limiting cases
- condition:
- v_f = v_i
- result:
- W_net = 0
- explanation:
- No change in speed means zero net work done.
- condition:
- v_i = 0
- result:
- W_net = ½mv_f²
- explanation:
- Starting from rest, net work equals final kinetic energy.
- condition:
- v_f = 0
- result:
- W_net = -½mv_i²
- explanation:
- Bringing to rest requires negative work (energy removed by friction/braking).
Context
Gaspard-Gustave de Coriolis · 1829
Coriolis formalized the concept of 'work' in Du calcul de l'effet des machines, linking force over distance to energy change.
Hook
A car brakes from 30 m/s to rest. How does braking force relate to stopping distance?
W_net = ΔKE = 0 - ½mv². For a 1500 kg car at 30 m/s: ΔKE = -675,000 J. With F_brake = 10,000 N, stopping distance d = 67.5 m.
Dimensions:
- lhs:
- W → [ML²T⁻²]
- rhs:
- ½m(v_f² - v_i²) → [M]·[LT⁻¹]² = [ML²T⁻²]
- check:
- Both sides are [ML²T⁻²] = Joule. ✓
Validity: Valid for non-relativistic speeds (v << 299792458 m/s) and point particles or rigid bodies. Does not account for internal energy changes (heat, deformation).