Introduction
Alright, the big one. Why does a sprinter need a few seconds to hit top speed, why does a heavier trolley accelerate less for the same push, why do crumple zones save lives, and why do your ears hurt at the deep end of a pool? Same chapter, all of it: measurement, forces, motion graphs, momentum, energy, and pressure.
This is your fast refresh, not the full textbook. One key formula per concept, one quick worked example, and the exact phrase the marker wants. Lock the formulas, read the graphs, walk in calm. Let's make the numbers move! 🏎️
Measurement and Scalars vs VectorsE1.1
You measure length with a ruler, volume by displacement in a measuring cylinder, and short times with a stopwatch (timing many swings of a pendulum, then dividing, beats timing one). A scalar has size only (mass, time, speed, energy); a vector has size and direction (force, velocity, acceleration, momentum). Two forces at an angle combine into a single resultant you find by scale drawing or, when they're perpendicular, by Pythagoras.
> 🔑 Key idea: Scalar = size only; vector = size + direction. Perpendicular vectors combine by $R = \sqrt{a^2 + b^2}$; use a scale diagram for other angles.
Worked Example: Two Pushes at a Right AngleE1.1
Problem A box is pushed with $3.0$ N east and $4.0$ N north at the same time. Find the size of the resultant force.
Step 1 The forces are perpendicular, so use Pythagoras: $R = \sqrt{3.0^2 + 4.0^2} = \sqrt{25} = 5.0$ N. (Direction would come from $\tan\theta = 4.0/3.0$, but the question only asked for the size.)
Speed, Velocity and AccelerationE1.2
Speed is distance per time; velocity is speed in a stated direction. Acceleration is how fast velocity changes: $a = \Delta v / \Delta t$. The exam loves speed-time graphs: the gradient is the acceleration and the area under the line is the distance travelled. A flat line means steady speed, a downward slope means deceleration. Near Earth, free-falling objects accelerate at $g \approx 9.8$ m/s² until air resistance grows to match weight, giving constant terminal velocity.
> 🔑 Key formulas: $a = \dfrac{\Delta v}{\Delta t}$. On a speed-time graph: gradient = acceleration, area = distance. Free-fall acceleration $g \approx 9.8$ m/s².
Worked Example: Acceleration from a Graph GradientE1.2
Problem A car speeds up from $8.0$ m/s to $20$ m/s in $6.0$ s. Find its acceleration, then the distance covered in that time.
Step 1 Acceleration is the gradient: $$a = \frac{\Delta v}{\Delta t} = \frac{20 - 8.0}{6.0} = 2.0 \text{ m/s}^2$$
Step 2 Distance is the area under the speed-time line: a rectangle ($8.0 \times 6.0 = 48$ m) plus a triangle ($\tfrac{1}{2} \times 6.0 \times 12 = 36$ m).
Reading both the gradient and the area off one graph is the single most common Motion exam move. 📈
$$s = 48 + 36 = 84 \text{ m}$$
Mass, Weight and Gravitational Field StrengthE1.3
Mass (kg) is the amount of matter and never changes. Weight (N) is the gravitational pull on that mass: $W = mg$, where $g$ is the gravitational field strength (≈ $9.8$ N/kg on Earth, less on the Moon). An astronaut in orbit isn't beyond gravity at all; they fall freely with their ship, so they feel weightless. Mass also measures inertia: the bigger the mass, the harder it is to start, stop, or turn.
> 🔑 Key formula: $W = mg$. Mass is constant everywhere; weight changes with $g$. On Earth $g \approx 9.8$ N/kg, on the Moon about $1.6$ N/kg.
Worked Example: Your Weight on MarsE1.3
Problem A rover has a mass of $180$ kg. Find its weight on Earth ($g = 9.8$ N/kg) and on Mars ($g = 3.7$ N/kg).
Step 1 Apply $W = mg$ twice: Earth $W = 180 \times 9.8 = 1764$ N $\approx 1.8$ kN; Mars $W = 180 \times 3.7 = 666$ N. Same mass, smaller field, smaller weight.
Density and FloatingE1.4
Density is mass packed into volume: $\rho = m/V$ (units kg/m³ or g/cm³). It decides floating: an object floats on a fluid if its density is less than the fluid's, and one liquid floats on another the same way (oil on water, water on syrup). Water is $1000$ kg/m³ ($1.0$ g/cm³), a handy reference. Find an irregular solid's volume by the rise in water level when you drop it in.
> 🔑 Key formula: $\rho = \dfrac{m}{V}$. Less dense floats on more dense. Water $= 1000$ kg/m³ $= 1.0$ g/cm³.
Worked Example: Will the Block Float?E1.4
Problem A plastic block has mass $240$ g and volume $300$ cm³. Will it float on water?
Step 1 Density $\rho = m/V = 240/300 = 0.80$ g/cm³. That's less than water's $1.0$ g/cm³, so it floats.
Forces, Hooke's Law and Newton's Second LawE1.5.1
A resultant (unbalanced) force changes motion; balanced forces leave it unchanged (Newton's first law). Newton's second law puts a number on it: $F = ma$. A spring obeys Hooke's Law, extension proportional to load, $F = kx$, up to its limit of proportionality. When the resultant force keeps pointing toward a centre, the object moves in a circle at constant speed but changing direction, so it's still accelerating (think a ball on a string, or a car rounding a bend).
> 🔑 Key formulas: $F = ma$ (resultant force, mass, acceleration) and $F = kx$ (spring force, spring constant, extension within the limit). Circular motion needs a centre-pointing resultant force.
Worked Example: Push a Trolley, Get a NumberE1.5.1
Problem A $1.5$ kg trolley is pushed with $9.0$ N forward against $3.0$ N of friction. Find its acceleration.
Step 1 First the resultant force: $F = 9.0 - 3.0 = 6.0$ N forward. Friction always opposes motion, so you subtract it.
Step 2 Now Newton's second law, rearranged for acceleration:
The golden habit: find the resultant force first, then divide by mass. Skip the resultant and you lose the marks. 🌀
$$a = \frac{F}{m} = \frac{6.0}{1.5} = 4.0 \text{ m/s}^2$$
Moments and EquilibriumE1.5.2
A moment is a turning effect: $M = F \times d$, force times the perpendicular distance from the pivot (units N·m). For an object that isn't turning, the principle of moments says total clockwise moments equal total anticlockwise moments about any pivot. Full equilibrium needs two conditions at once: no resultant force and no resultant moment. This is how see-saws, cranes, and spanners are analysed.
> 🔑 Key formula: Moment $M = F \times d$. Balanced: clockwise moments $=$ anticlockwise moments. Equilibrium needs zero resultant force and zero resultant moment.
Worked Example: Find the Missing DistanceE1.5.2
Problem A $4$ N weight sits $3.0$ m left of a pivot. Where must a $6$ N weight hang on the right to balance the beam?
Step 1 Apply the principle of moments, anticlockwise = clockwise: $4 \times 3.0 = 6 \times d$, so $d = 12/6 = 2.0$ m from the pivot.
Centre of Gravity and StabilityE1.5.3
The centre of gravity is the single point where an object's whole weight seems to act. An object is stable when this point sits low and over a wide base, and it tips over only when its centre of gravity passes outside the base. That's why racing cars are low and wide and a tall thin glass topples easily. You find the centre of gravity of a flat card by hanging it from two points and marking the plumb-line each time; the lines cross at the centre of gravity.
> 🔑 Key idea: Low centre of gravity + wide base = stable. It topples once the line of action of the weight falls outside the base.
Worked Example: The Plumb-Line TrickE1.5.3
Problem Describe how to find the centre of gravity of an irregular flat card.
Step 1 Hang the card freely from a pin near one edge and hang a plumb-line from the same pin; mark the vertical line on the card. Repeat from a second point. The centre of gravity is where the two lines cross, because a hanging object always settles with its centre of gravity directly below the pivot.
Momentum and ImpulseE1.6
Momentum is mass in motion: $p = mv$ (units kg·m/s), a vector. In any collision with no outside forces, total momentum is conserved: the total before equals the total after. Impulse is the change in momentum, $Ft = \Delta(mv)$, and it explains safety design: a crumple zone or airbag makes a collision last longer, so for the same momentum change the force is smaller. Same physics behind bending your knees when you land.
> 🔑 Key formulas: $p = mv$; total momentum conserved in collisions. Impulse $Ft = \Delta(mv)$: longer impact time $\Rightarrow$ smaller force.
Worked Example: Sticky Crash, Same Total pE1.6
Problem A $1000$ kg car at $20$ m/s hits a stationary $1000$ kg car and they lock together. Find their common speed afterwards.
Step 1 Momentum before: $p = (1000)(20) + (1000)(0) = 20\,000$ kg·m/s.
Step 2 After, the combined $2000$ kg moves at $v$, and momentum is conserved:
Double the moving mass, half the speed: total momentum unchanged. 💥
$$2000\,v = 20\,000 \quad\Rightarrow\quad v = 10 \text{ m/s}$$
Energy Stores, Conservation and Sankey DiagramsE1.7.1
Energy is stored (kinetic, gravitational, elastic, chemical, nuclear, thermal) and transferred between stores, never created or destroyed: that's conservation of energy. Two formulas dominate: kinetic energy $E_k = \tfrac{1}{2}mv^2$ and change in gravitational PE $\Delta E_p = mg\Delta h$. A falling object swaps PE for KE; ignoring air resistance, every joule of PE lost becomes KE. Sankey diagrams draw the flow as arrows whose width shows how much energy goes to useful output versus wasted heat.
> 🔑 Key formulas: $E_k = \tfrac{1}{2}mv^2$ and $\Delta E_p = mg\Delta h$. Energy is conserved: PE lost $=$ KE gained (no air resistance). Sankey widths $\propto$ energy.
Worked Example: Drop a Ball, Predict the SpeedE1.7.1
Problem A $0.50$ kg ball is dropped from $1.8$ m. Find its speed just before it lands (take $g = 10$ N/kg, ignore air resistance).
Step 1 All the gravitational PE becomes kinetic energy: $\Delta E_p = mg\Delta h = 0.50 \times 10 \times 1.8 = 9.0$ J.
Step 2 Set $E_k = 9.0$ J and solve for $v$:
Notice the mass cancels out of the final speed: every object falls to the same speed from the same height (no air resistance). 🏀
$$\tfrac{1}{2}mv^2 = 9.0 \;\Rightarrow\; v^2 = \frac{2 \times 9.0}{0.50} = 36 \;\Rightarrow\; v = 6.0 \text{ m/s}$$
Work DoneE1.7.2
Work done is energy transferred by a force: $W = Fd$, force times the distance moved in the direction of the force (units: joules, the same unit as energy). Lift a box and you do work against gravity that becomes gravitational PE; push a crate across a floor and your work becomes heat in the friction. No movement means no work, however hard you strain.
> 🔑 Key formula: $W = Fd$ (force × distance moved in the force's direction). Work done = energy transferred, measured in joules.
Worked Example: Loading the Top ShelfE1.7.2
Problem You lift a $12$ N box through a height of $1.5$ m onto a shelf. How much work do you do, and where does that energy go?
Step 1 $W = Fd = 12 \times 1.5 = 18$ J. That energy is now stored as the box's extra gravitational PE.
Energy ResourcesE1.7.3
We generate electricity from renewables (solar, wind, hydro, geothermal, tidal, biomass) and non-renewables (coal, oil, gas, nuclear). Most trace back to the Sun: fossil fuels are ancient stored sunlight, wind and waves are driven by solar heating; the exceptions are nuclear, geothermal, and tidal (the Moon's pull). The Sun's own energy comes from nuclear fusion of hydrogen into helium. Exam answers weigh reliability, cost, and pollution against each other.
> 🔑 Key idea: Renewable vs non-renewable. Most resources trace to the Sun; nuclear, geothermal, and tidal don't. The Sun is powered by hydrogen fusion.
Worked Example: Which Power Station Wastes Less?E1.7.3
Problem Station A turns $40\%$ of its fuel energy into electricity; Station B turns $35\%$. For every $100$ MJ of fuel, how much useful electrical energy does each give, and which wastes less?
Step 1 A gives $0.40 \times 100 = 40$ MJ; B gives $0.35 \times 100 = 35$ MJ. Station A delivers more useful energy per unit of fuel, so it wastes less.
Power and EfficiencyE1.7.4
Power is how fast energy is transferred (or work is done): $P = E/t = W/t$, measured in watts (1 W = 1 J/s). Efficiency is the fraction of input energy that comes out useful: $\text{efficiency} = \dfrac{\text{useful output}}{\text{total input}} \times 100\%$. No real machine reaches $100\%$ because some energy always ends up as wasted heat (friction, sound). A more powerful motor does the same job faster; a more efficient one wastes less doing it.
> 🔑 Key formulas: $P = \dfrac{E}{t}$ (watts). Efficiency $= \dfrac{\text{useful energy out}}{\text{total energy in}} \times 100\%$, always under $100\%$.
Worked Example: How Much Hits the Hook?E1.7.4
Problem An electric motor takes in $500$ J of electrical energy and lifts a load, giving it $300$ J of gravitational PE. Find its efficiency.
Step 1 Efficiency $= \dfrac{\text{useful out}}{\text{total in}} \times 100\% = \dfrac{300}{500} \times 100\% = 60\%$. The missing $200$ J became heat and sound.
PressureE1.8
Pressure is force spread over area: $p = F/A$ (units: pascals, $1$ Pa $= 1$ N/m²). A sharp knife or a stiletto heel concentrates force on a tiny area for huge pressure; a snowshoe spreads it out for tiny pressure. In a fluid, pressure increases with depth: $\Delta p = \rho g \Delta h$. That's why a dam is thicker at the bottom and why your ears hurt as you dive deeper, with atmospheric pressure pushing on top of the water's.
> 🔑 Key formulas: $p = \dfrac{F}{A}$ (pascals). In a fluid, pressure rises with depth: $\Delta p = \rho g \Delta h$. Small area $\Rightarrow$ big pressure.
Worked Example: Pressure at the Deep EndE1.8
Problem Find the extra water pressure on a swimmer $2.5$ m below the surface. Take $\rho_\text{water} = 1000$ kg/m³ and $g = 10$ N/kg.
Step 1 Use $\Delta p = \rho g \Delta h = 1000 \times 10 \times 2.5 = 25\,000$ Pa $= 25$ kPa. That's on top of atmospheric pressure, which is why the deep end squeezes your ears.
Key Points to Remember
- Scalars have size only; vectors (force, velocity, acceleration, momentum) have size and direction. Perpendicular vectors add by Pythagoras.
- On a speed-time graph, gradient = acceleration and area = distance. $a = \Delta v/\Delta t$; free-fall $g \approx 9.8$ m/s².
- $W = mg$ (weight vs constant mass); $\rho = m/V$ (less dense floats); $F = ma$ (resultant force first); $F = kx$ (Hooke's Law).
- Moments: $M = Fd$, balanced when clockwise = anticlockwise. Stability needs a low centre of gravity over a wide base.
- Momentum $p = mv$ is conserved in collisions; longer impact time means smaller force (crumple zones). $E_k = \tfrac{1}{2}mv^2$, $\Delta E_p = mg\Delta h$, energy always conserved.
- $W = Fd$; $P = E/t$ (watts); efficiency = useful/total × 100% (never 100%); $p = F/A$ and $\Delta p = \rho g \Delta h$ in fluids.