Playground
Ball rolls across screen with speed controlled by slider. Energy bar and v² scaling are visualized in real time.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Kinetic Energyoutput Translational kinetic energy of the object | J | ML²T⁻² | 0 – 50000 | |
| Mass Mass of the moving object | kg | M | 0.01 – 1000 | |
| Velocity Speed of the object | m/s | LT⁻¹ | 0 – 300 |
Deep dive
Derivation
Integrate F·dx = ma·dx = m(dv/dt)dx = mv·dv from 0 to v. Result: W = ½mv² - 0 = ½mv².
Experimental verification
Ballistic pendulum experiments, particle accelerator energy measurements, vehicle crash energy analysis.
Common misconceptions
- Kinetic energy is not a vector — it's always positive
- Doubling speed quadruples KE, not doubles it
- KE depends on the reference frame — there is no absolute kinetic energy
Real-world applications
- Vehicle braking distance calculations
- Ballistics and impact analysis
- Wind turbine power output
- Roller coaster design
Worked examples
Baseball vs bowling ball energy
Given:
- m_ball:
- 0.145
- v_ball:
- 40
- m_bowl:
- 7.26
- v_bowl:
- 7
Find: KE comparison
Solution
Baseball: ½(0.145)(40²) = 116 J. Bowling ball: ½(7.26)(7²) = 177.9 J. Bowling ball wins.
Braking distance at double speed
Given:
- m:
- 1500
- v1:
- 15
- v2:
- 30
Find: KE ratio
Solution
KE at 30 m/s = 4 × KE at 15 m/s. Braking distance quadruples from 16.9 m to 67.5 m.
Scenarios
What if…
- scenario:
- What if speed doubles?
- answer:
- KE quadruples (v² dependence). This is why highway crashes at 120 km/h are 4× more energetic than at 60 km/h.
- scenario:
- What if measured from a moving train?
- answer:
- KE depends on reference frame. A ball at rest on a train has KE = 0 for a passenger but KE = ½mv² for a ground observer.
- scenario:
- What about rotating objects?
- answer:
- Add rotational KE = ½Iω². A rolling ball has both translational and rotational KE.
Limiting cases
- condition:
- v → 0
- result:
- KE → 0
- explanation:
- A stationary object has no kinetic energy.
- condition:
- v → c
- result:
- Use relativistic KE
- explanation:
- At speeds approaching light, KE = (γ-1)mc² replaces the classical formula.
- condition:
- m → 0
- result:
- KE → 0
- explanation:
- Massless classical particles carry no kinetic energy (photons use E=pc).
Context
Gottfried Wilhelm Leibniz · 1689
Leibniz argued that 'vis viva' (living force, mv²) was conserved in collisions, correcting Descartes' mv.
Hook
A 0.145 kg baseball at 40 m/s vs a 7.26 kg bowling ball at 7 m/s — which packs more energy?
Compute KE = ½mv² for both objects. The baseball has 116 J while the bowling ball has 178 J — mass wins despite lower speed.
Dimensions:
- lhs:
- KE → [ML²T⁻²]
- rhs:
- ½·m·v² → [M]·[LT⁻¹]² = [ML²T⁻²]
- check:
- Both sides are [ML²T⁻²] = Joule. ✓
Validity: Valid for non-relativistic speeds (v << 299792458 m/s). For v approaching c, use relativistic kinetic energy KE = (γ-1)mc².