Motion, Force, Gravitation, Work and Energy
Equations of motion and free fall
For uniform acceleration a, with initial velocity u, final velocity v, time t and displacement s: v = u + at, s = ut + ½at², v² = u² + 2as. Pick the equation that omits the quantity you do not have: no s gives the first, no v the second, no t the third.
Worked numbers. A car from rest with a = 2 m/s² for 10 s reaches v = 20 m/s and covers s = ½ × 2 × 100 = 100 m. If it then brakes to rest in 40 m, 0 = 20² + 2a(40) gives a = −400/80 = −5 m/s², and the stopping time is 20/5 = 4 s.
Free fall is motion under gravity alone: a = g = 9.8 m/s² downwards, the same for every mass in a vacuum. Dropped from 19.6 m: t = √(2 × 19.6 / 9.8) = 2 s and v = 9.8 × 2 = 19.6 m/s. Thrown upwards at 19.6 m/s: the speed falls to zero after 2 s, the greatest height is 19.6² / (2 × 9.8) = 19.6 m, and the total time of flight back to the thrower is 4 s. At the top the velocity is zero but the acceleration is still 9.8 m/s² downwards.
Newton's laws and momentum
- First law (inertia): a body stays at rest or in uniform motion in a straight line unless an unbalanced force acts. A passenger lurches forward when a bus brakes suddenly because the body tends to keep moving.
- Second law: the rate of change of momentum is proportional to the applied force, so F = ma. Momentum p = mv, unit kg m/s. 1 newton = 1 kg m/s².
- Third law: to every action there is an equal and opposite reaction. The two forces act on different bodies, so they never cancel each other.
Worked numbers. A 6 kg body under a net force of 30 N has a = 30/6 = 5 m/s². A 2 kg body moving at 4 m/s pushed forward by 6 N for 5 s has a = 3 m/s², so v = 4 + 3 × 5 = 19 m/s. A 0.5 kg ball at 10 m/s stopped in 0.1 s: force = 0.5 × 10 / 0.1 = 50 N. The product of force and time, Ft = change in momentum, is called impulse; a longer stopping time (cricket catch with the hands drawn back) means a smaller force.
Conservation of momentum: with no external force, total momentum before = after. A 2 kg ball at 3 m/s hits and sticks to a 1 kg ball at rest: 6 = 3v, so v = 2 m/s. Recoil: a gun of 4 kg fires a 0.02 kg bullet at 300 m/s; 0 = 0.02 × 300 + 4 × V gives V = −1.5 m/s, that is 1.5 m/s backwards.
Universal gravitation, g and weight
Newton's law: every body attracts every other with a force F = G m₁m₂ / r², where r is the distance between the centres. G = 6.67 × 10⁻¹¹ N m²/kg² is the universal gravitational constant (measured by Cavendish). The force is inverse-square: doubling r makes F one-fourth; tripling r makes it one-ninth. Example: two 10 kg masses 1 m apart attract with 6.67 × 10⁻¹¹ × 100 / 1 = 6.67 × 10⁻⁹ N, a force too small to feel.
g = GM / R² is the acceleration due to gravity at the surface of a planet of mass M and radius R; for Earth, g ≈ 9.8 m/s². It does not depend on the falling body's mass. It is slightly larger at the poles than at the equator (Earth is flattened and rotates) and decreases with height above the surface and with depth below it. Weight W = mg is the force of gravity: a 60 kg person weighs 60 × 9.8 = 588 N on Earth. On the Moon g is about one-sixth of Earth's, so the weight is 588/6 = 98 N, while the mass stays 60 kg.
Scaling. Since g ∝ M/R², a planet with twice Earth's mass and the same radius has g = 19.6 m/s²; with twice the mass and twice the radius, g = 2/4 × 9.8 = 4.9 m/s². Escape velocity from Earth is about 11.2 km/s, and the orbital speed of a satellite close to the surface is about 7.9 km/s. A satellite in orbit is in free fall, so astronauts inside feel weightless although gravity still acts.
Work, energy and power
Work W = F s cos θ, in joules (1 J = 1 N × 1 m). Work is zero when there is no displacement, or when the force is perpendicular to the motion: a porter walking level with a load on the head does no work against gravity. Kinetic energy KE = ½mv²; potential energy (near the ground) PE = mgh. The work-energy theorem says net work equals the change in kinetic energy.
Worked numbers. A 2 kg body at 10 m has PE = 2 × 9.8 × 10 = 196 J. A 1000 kg car at 72 km/h (20 m/s) has KE = ½ × 1000 × 400 = 200 000 J = 200 kJ. Doubling the speed multiplies KE by 4. A ball dropped from 19.6 m reaches the ground at v = √(2 × 9.8 × 19.6) = 19.6 m/s: its PE has turned entirely into KE, the law of conservation of energy (energy changes form but the total stays constant).
Power P = W / t, in watts (1 W = 1 J/s); 1 horse power = 746 W. A pump lifting 500 kg of water through 20 m in 100 s does W = 500 × 9.8 × 20 = 98 000 J, so P = 980 W. Commercial unit: 1 kWh = 3.6 × 10⁶ J; a 100 W bulb on for 10 hours uses 1 kWh, one "unit" of electricity.
| Device | Energy change |
|---|---|
| Dynamo (generator) | Mechanical to electrical |
| Electric motor | Electrical to mechanical |
| Microphone | Sound to electrical |
| Solar cell | Light to electrical |
Pressure, buoyancy and Archimedes' principle
Pressure = force / area, unit pascal (1 Pa = 1 N/m²). A 600 N person standing on 0.03 m² of shoe sole exerts 600 / 0.03 = 20 000 Pa; a sharp knife cuts easily because the same force acts on a tiny area. In a liquid, pressure at depth h is P = hρg: at 10 m in water (ρ = 1000 kg/m³) it is 10 × 1000 × 9.8 = 98 000 Pa above the surface pressure. Atmospheric pressure is about 1.013 × 10⁵ Pa, equal to a 76 cm column of mercury in a barometer.
Pascal's law: pressure applied to an enclosed fluid is transmitted undiminished everywhere in it. A hydraulic lift therefore multiplies force in the ratio of the piston areas: 100 N on a 5 cm² piston gives F = 100 × 500 / 5 = 10 000 N on a 500 cm² piston (work is not multiplied: the small piston moves 100 times farther).
Archimedes' principle: a body wholly or partly immersed in a fluid feels an upward buoyant force equal to the weight of the fluid it displaces. A 500 cm³ object fully under water displaces 0.5 kg of water, so the buoyant force is 0.5 × 9.8 = 4.9 N. A body weighing 10 N in air and 6 N in water has buoyant force 4 N, so its relative density is 10 / 4 = 2.5. A body floats if its density is less than the liquid's, and sinks if greater; a floating body displaces fluid equal to its own weight. Ice (about 0.92 times the density of water) floats with roughly nine-tenths submerged. A ship floats because its hollow shape displaces much water; a hydrometer measures liquid density by the same principle.
Key takeaways
- Distance and speed are scalars; displacement and velocity are vectors. Equal-distance average speed is the harmonic mean (60 and 40 give 48). Convert km/h to m/s by × 5/18.
- Slope of v-t is acceleration, area under v-t is displacement. Use v = u + at, s = ut + ½at², v² = u² + 2as; free fall has a = 9.8 m/s² for every mass.
- F = ma, p = mv, impulse = Ft. Action and reaction are equal, opposite and on different bodies. Total momentum is conserved when no external force acts (recoil).
- F = Gm₁m₂/r² is inverse-square; g = GM/R². Mass is constant, weight W = mg changes: Moon weight is about one-sixth. G is universal, g is local.
- W = Fs cos θ (zero if perpendicular), KE = ½mv² (doubling speed gives four times), PE = mgh, P = W/t, 1 hp = 746 W, 1 kWh = 3.6 × 10⁶ J.
- P = F/A and P = hρg; hydraulic lift multiplies force by the area ratio. Buoyant force equals the weight of fluid displaced; a body floats if its density is less than the liquid's.
Practice questions (18)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
A runner completes exactly one lap of a 400 m circular track in 80 s and stops at the starting point. What are the runner's average speed and average velocity for the lap?
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Answer: C — 5 m/s and 0 m/s
Average speed is distance over time: 400 / 80 = 5 m/s. Displacement is zero because start and end coincide, so average velocity is 0. The tempting 5 m/s for both treats velocity as if it were speed; speed is never zero here because the runner moved.A train runs at 72 km/h. Its speed in metres per second is
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Answer: A — 20 m/s
Multiply km/h by 5/18: 72 × 5/18 = 20 m/s. The value 259.2 comes from wrongly multiplying by 18/5, which is the opposite conversion (m/s to km/h).A car covers half of a journey at 60 km/h and the other half, in distance, at 40 km/h. Its average speed for the whole journey is
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Answer: B — 48 km/h
For equal distances, average speed = total distance / total time = 2 × 60 × 40 / (60 + 40) = 48 km/h. The simple mean 50 km/h is right only when the two speeds are kept for equal times, which is not the case here.A body's speed rises uniformly from 0 to 10 m/s in 5 s, stays at 10 m/s for 10 s, then falls uniformly to 0 in 5 s. The total distance travelled is
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Answer: D — 150 m
Distance is the area under the velocity-time graph: 25 m while speeding up, 100 m at steady speed, 25 m while slowing, total 150 m. The figure 200 m comes from taking the full 20 s at 10 m/s and ignoring the triangular ends.A stone is thrown vertically upwards at 19.6 m/s from the ground. Taking g = 9.8 m/s² and ignoring air resistance, the total time it takes to return to the ground is
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Answer: A — 4 s
Time to the top is u/g = 19.6 / 9.8 = 2 s, and the fall takes the same time, so the total is 4 s. The value 2 s is only the rising time; the greatest height 19.6 m is reached at that moment.A standing passenger lurches forward when a moving bus brakes suddenly. This is best explained by
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Answer: B — Newton's first law, the inertia of the passenger's body
The feet stop with the bus but the upper body tends to continue in its state of motion; that tendency is inertia, the content of the first law. The second law relates force and acceleration of one body and does not by itself explain this tendency.A 2 kg body moving at 4 m/s along a smooth line is pushed forward by a constant force of 6 N for 5 s. Its final speed is
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Answer: C — 19 m/s
The acceleration is F/m = 6/2 = 3 m/s², so v = u + at = 4 + 3 × 5 = 19 m/s. The value 15 m/s is what you get by forgetting the initial 4 m/s, and 34 m/s results from using the force 6 N as if it were the acceleration.A gun of mass 4 kg fires a bullet of mass 20 g at 300 m/s. The speed with which the gun recoils is
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Answer: D — 1.5 m/s
By conservation of momentum, 4 × V = 0.02 × 300, so V = 6/4 = 1.5 m/s. The answer 15 m/s comes from using 20 g as 0.2 kg; the bullet mass must be converted to 0.02 kg. The value 6 m/s is the bullet's momentum left undivided by the gun's mass.Judge these statements. P: The mass of a body is the same on the Moon as on the Earth. Q: Action and reaction forces act on the same body and cancel. R: Momentum is a vector quantity. S: The SI unit of force is kg m/s. Which are correct?
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Answer: B — P and R only
Mass is constant everywhere (P is true) and momentum has the direction of velocity (R is true). Action and reaction act on different bodies, so Q is false. The unit of force is the newton, kg m/s², not kg m/s, which is the unit of momentum, so S is false.The SI unit of the universal gravitational constant G is
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Answer: A — N m² kg⁻²
From F = G m₁m₂ / r², G = F r² / (m₁m₂), whose unit is N m² / kg². The unit m/s² belongs to g, not G, which is the commonly confused pair.The distance between the centres of two bodies is tripled while their masses stay the same. The gravitational force between them becomes
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Answer: C — one-ninth of the original
Force varies as 1/r², so tripling r divides the force by 3² = 9. One-third would be the result for an inverse-first-power law, which gravitation does not follow.A planet has twice the mass and twice the radius of the Earth. If g on Earth is 9.8 m/s², g on that planet is
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Answer: D — 4.9 m/s²
Since g = GM/R², doubling M doubles g but doubling R divides it by 4, so g = 9.8 × 2/4 = 4.9 m/s². The value 19.6 counts only the mass, and 9.8 assumes the two changes cancel, which they do not.A person has a mass of 60 kg on the Earth. Taking g = 9.8 m/s² on the Earth and the Moon's g as one-sixth of that, the person's weight on the Moon is
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Answer: B — 98 N
Weight on the Moon = m × g/6 = 60 × 9.8/6 = 98 N. The value 588 N is the Earth weight, and 10 N would be the Moon mass in kilograms divided by six, mixing mass with weight; the mass stays 60 kg.A car of mass 1000 kg moves at 72 km/h. Its kinetic energy is
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Answer: A — 200 kJ
Convert first: 72 km/h = 20 m/s. KE = ½ × 1000 × 20² = 200 000 J = 200 kJ. Using 72 directly gives 2.6 million joules, and 400 kJ forgets the factor ½.A pump raises 500 kg of water through a height of 20 m in 100 s. Taking g = 9.8 m/s², the power it delivers to the water is
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Answer: B — 980 W
Work = mgh = 500 × 9.8 × 20 = 98 000 J; power = 98 000 / 100 = 980 W. The value 9800 W comes from dividing the work by 10 s instead of 100 s, and 1000 W comes from the rough g = 10, which the question did not allow.Match the quantity with its SI unit. P: work, Q: power, R: pressure, S: momentum. Units are the joule, the watt, the pascal and kg m/s. Choose the correct pairing.
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Answer: D — P joule, Q watt, R pascal, S kg m/s
Work is in joules, power in watts (J/s), pressure in pascals (N/m²) and momentum in kg m/s. The pairing that swaps work and power is the usual trap, since the joule and the watt differ by a factor of time.In a hydraulic lift the small piston has area 5 cm² and the large piston 500 cm². A force of 100 N on the small piston can balance a force on the large piston of
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Answer: C — 10 000 N
By Pascal's law the pressure is the same on both: 100/5 = F/500, so F = 10 000 N. The force is multiplied by the area ratio 100, but the work is not, because the small piston moves 100 times farther.A metal piece weighs 10 N in air and 6 N when fully immersed in water. Its relative density (density relative to water) is
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Answer: D — 2.5
The buoyant force is the loss of weight, 10 − 6 = 4 N. Relative density = weight in air / loss of weight = 10 / 4 = 2.5. The value 1.67 divides by the weight in water instead of by the loss, which is the common slip.