Worked examples

A cyclist and her bicycle have a combined mass of $80\,\text{kg}$. She speeds up from $4.0\,\text{m/s}$ to $9.0\,\text{m/s}$. Calculate the increase in the energy in the kinetic energy store of the cyclist and bicycle.

Find the kinetic store at each speed, then subtract — never square the change in speed.

A drone of mass $1200\,\text{g}$ rises vertically and gains $330\,\text{J}$ of energy in its gravitational potential energy store. Gravitational field strength $= 9.8\,\text{N/kg}$. Calculate the height the drone rises. Give your answer to 2 significant figures.

Convert grams to kilograms, rearrange Ep = mgh for the height, and round once at the end.

In the specific heat capacity required practical, a student heats a $1.0\,\text{kg}$ aluminium block with a $50\,\text{W}$ heater for $6.0$ minutes. The temperature of the block rises from $21.0\,{}^\circ\text{C}$ to $40.5\,{}^\circ\text{C}$. (a) Calculate the specific heat capacity of aluminium given by these results. (b) The accepted value is about $900\,\text{J/kg}\,{}^\circ\text{C}$. Explain why the student's value is higher.

Energy from E = Pt, the temperature change, then c = ΔE ÷ (mΔθ) — and why the result comes out too high.

An electric motor has an input power of $400\,\text{W}$. In $6.0\,\text{s}$ it transfers $1800\,\text{J}$ of energy usefully to lift a load. Calculate (a) the total energy supplied to the motor, (b) the efficiency of the motor, (c) the useful power output of the motor.

Input energy from power × time, efficiency as useful ÷ total, and the useful power output — with a check that efficiency is below 1.

A lift motor raises a lift of mass $540\,\text{kg}$ at a steady speed through a vertical height of $24\,\text{m}$ in $18\,\text{s}$. Gravitational field strength $= 9.8\,\text{N/kg}$. Calculate the useful power output of the motor. Give your answer in kilowatts to 2 significant figures.

Step 1 finds the energy transferred, step 2 divides by the time — carrying the unrounded energy through.

A $4.0\,\Omega$ resistor and an $8.0\,\Omega$ resistor are connected in series to a $12\,\text{V}$ battery. (a) Calculate the current in the circuit and the potential difference across the $8.0\,\Omega$ resistor. (b) The two resistors are then reconnected in parallel across the same battery. Calculate the current from the battery and explain why the total resistance is now less than $4.0\,\Omega$.

The same two resistors in series, then in parallel: share the p.d., add the branch currents, and explain why parallel resistance falls.

A student finds the density of an irregular stone. Its mass is $84\,\text{g}$. When it is lowered into a full displacement can, $30\,\text{cm}^3$ of water flows out into a measuring cylinder. Calculate the density of the stone in $\text{kg/m}^3$.

Volume by displacement, then the unit conversions that decide the mark: g to kg and cm³ to m³.

A uniform seesaw is pivoted at its centre. Child A, weight $320\,\text{N}$, sits $1.5\,\text{m}$ to the left of the pivot, and a bag of weight $80\,\text{N}$ rests $0.50\,\text{m}$ to the left of the pivot. Child B, weight $400\,\text{N}$, sits on the right. How far from the pivot must Child B sit for the seesaw to balance?

Total anticlockwise moment = total clockwise moment, with every force paired with its own distance.

A diver is $12\,\text{m}$ below the surface of the sea. The density of sea water is $1030\,\text{kg/m}^3$ and gravitational field strength is $9.8\,\text{N/kg}$. (a) Calculate the pressure due to the water at this depth. (b) The glass window of the diver's mask has an area of $0.015\,\text{m}^2$. Calculate the force on the window due to the water.

Pressure from a liquid column with p = hρg, then the force on a surface with F = pA.

A trolley of mass $0.80\,\text{kg}$ moving at $2.5\,\text{m/s}$ collides with a stationary trolley of mass $1.2\,\text{kg}$. The trolleys stick together. (a) Calculate their velocity just after the collision. (b) The collision lasts $0.040\,\text{s}$. Calculate the average force on the $1.2\,\text{kg}$ trolley.

Conservation of momentum for the velocity after, then force as the rate of change of momentum.

Water waves are made in a ripple tank. $12$ waves pass a fixed point in $8.0\,\text{s}$, and the distance between neighbouring crests is $40\,\text{cm}$. (a) Calculate the wave speed. (b) The waves pass into shallower water, where they travel at $0.45\,\text{m/s}$. Their frequency does not change. Calculate the new wavelength.

Frequency from a count, wavelength in metres, v = fλ — then a new wavelength when only the speed changes.

A ray of light in air strikes one face of a rectangular glass block at an angle of incidence of $40^\circ$. It passes through the block and leaves through the opposite, parallel face. Describe how to draw the ray diagram and the path of the ray. Higher tier: explain why the ray changes direction at the first face.

Describe the ray diagram step by step — normals, angles from the normal, the bend in and out — then, for Higher tier, explain it with wave speed.