Conservation of Mechanical Energy
Senior 2 · Physics
Conservation of Mechanical Energy
Lesson 2 of 4
20 min
A moto-taxi racing down the hill from Kimihurura, a 20-litre jerrycan lifted onto a bicycle carrier, and water held high behind a dam all have something in common: each can do work. The moto can do it because it is moving; the jerrycan and the dam water can do it because of where they are. Physicists call these two kinds of energy kinetic energy and potential energy.
Energy is the ability to do work, and it is measured in joules (J). Road-safety officers, engineers at hydropower stations and builders all use the two formulas in this lesson.
A moto and its rider have a total mass of 150 kg and travel at 12 m/s (about 43 km/h).
KE = ½mv² = ½ × 150 × 12² = ½ × 150 × 144 = 10 800 J (10.8 kJ).
Now the moto speeds up to 24 m/s, twice as fast:
KE = ½ × 150 × 24² = ½ × 150 × 576 = 43 200 J.
Doubling the speed made the kinetic energy four times bigger, because the speed is squared. This is why a crash at twice the speed is so much more dangerous: the brakes and the road must remove four times as much energy.
Kinetic energy is the energy an object has because it is moving:
KE = ½mv²
where m is the mass in kilograms (kg), v is the speed in metres per second (m/s), and KE is in joules. One joule is 1 kg·m²/s², the same as 1 newton-metre (N·m). A force must do work on an object to give it kinetic energy: the work done by the net force equals the gain in kinetic energy.
Uwase lifts a 20 kg jerrycan of water steadily through 1.5 m onto a bicycle carrier. Take g = 9.8 m/s².
To lift it at a steady speed she must push up with a force equal to its weight: F = mg = 20 × 9.8 = 196 N.
Work done = force × distance = 196 × 1.5 = 294 J.
That work is not lost. It is stored as gravitational potential energy of the jerrycan–Earth system, and it comes back out if the jerrycan falls.
Gravitational potential energy (PE) is the energy a system has because of the height of an object above a chosen level:
PE = mgh
where m is in kg, g is the gravitational field strength (9.8 N/kg, the same number as 9.8 m/s²) and h is the height in metres above the reference level.
You choose the reference level, usually the floor or the ground. Only changes in potential energy have physical meaning. Example: a 2 kg book sits on a shelf 1.2 m above a table, and the table top is 0.8 m above the floor.
Both answers are correct for their own reference level. If the book falls from the shelf onto the table, it loses 23.52 J whichever level you chose.
A stretched catapult rubber, a compressed spring in a mattress and a bent bow also store energy, because they have been deformed (stretched, squeezed or bent). This is elastic potential energy. Let go, and it becomes kinetic energy of the stone or the arrow.
The mechanical energy (ME) of a system is the sum of its kinetic and potential energy: ME = KE + PE.
Worked example: a 0.5 kg ball is 3.0 m above the ground and moving at 4.0 m/s. PE = 0.5 × 9.8 × 3.0 = 14.7 J. KE = ½ × 0.5 × 4.0² = 4.0 J. ME = 14.7 + 4.0 = 18.7 J.
Tip
Square first, then halve. The most common error in KE = ½mv² is squaring the wrong thing. Work out v² first, then multiply by m, then halve.
Check the effect before you calculate: double v → 4 × KE; double m → 2 × KE; double h → 2 × PE.
Units: use kg, m and m/s only. Convert km/h to m/s by dividing by 3.6, and grams to kilograms by dividing by 1000.
Take g = 9.8 m/s².
A car of mass 1200 kg travels at 72 km/h on the Kigali–Musanze road. Find its kinetic energy.
Solution: v = 72 ÷ 3.6 = 20 m/s. KE = ½ × 1200 × 20² = ½ × 1200 × 400 = 240 000 J (240 kJ).
A 60 kg student climbs the stairs to a classroom 3.0 m higher. How much gravitational potential energy does she gain?
Solution: PE = mgh = 60 × 9.8 × 3.0 = 1764 J.
A 0.2 kg ball has 10 J of kinetic energy. How fast is it moving?
Solution: Rearrange KE = ½mv²: v = √(2 × KE ÷ m) = √(2 × 10 ÷ 0.2) = √100 = 10 m/s.
At what height does a 5.0 kg sack of maize have 245 J of potential energy relative to the ground?
Solution: h = PE ÷ (mg) = 245 ÷ (5.0 × 9.8) = 245 ÷ 49 = 5.0 m.
Bicycle A (with rider, 20 kg in total, a child's bike) moves at 6 m/s. Bicycle B (40 kg in total) moves at 3 m/s. Which has more kinetic energy?
Solution: KE of A = ½ × 20 × 6² = 360 J. KE of B = ½ × 40 × 3² = 180 J. A has twice as much, even though it has half the mass, because speed is squared.
Remember
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