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hscscience Maths Std · Y11
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Module 2 · L18 of 22 ~55 min ⚡ +95 XP available

Units of Energy and Mass

From the joules in a lightning bolt to the kilojoules on a food label, energy and mass have a hierarchy of units. Convert fluently between them and you can answer any real-world measurement question.

Today's hook, Your household electricity bill charges you by the kilowatt-hour (kWh). A standard electric kettle is rated at 2400 W. If you boil it for 3 minutes, how much energy does it use, and roughly what does that cost at 30 cents per kWh?
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Recall, your gut answer first

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

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Recall, your gut answer first
+5 XP warm-up

Your household electricity bill charges you by the kilowatt-hour (kWh). A standard electric kettle is rated at 2400 W. If you boil the kettle for 3 minutes, how much energy does it use, and roughly what does that cost if electricity is 30 cents per kWh?

Make a rough guess before the lesson teaches you the method.

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The unit hierarchies, mass and energy

+5 XP to read Mass and energy both have a hierarchy of units each step multiplies or divides by a fixed factor.

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The unit hierarchies, mass and energy
+5 XP to read

Mass and energy both have a hierarchy of units each step multiplies or divides by a fixed factor. Identify the direction first: going to a smaller unit means multiply; going to a larger unit means divide.

Mass chain: mg → g → kg → t, each step ×1000 going down or ÷1000 going up.

Energy chain: J → kJ → MJ (÷1000 each step). Food uses kcal: 1 kcal = 4.184 kJ. Electricity uses kWh: 1 kWh = 3600 kJ.

Conversion chains for mass and energy units.
$E \text{ (kWh)} = P \text{ (kW)} \times t \text{ (h)}$
Mass units
mg → g → kg → t, each step ÷1000 going up. To go to a smaller unit: multiply. Going to a larger unit: divide.
Energy units
J → kJ (÷1000). kJ → kcal (÷4.184). 1 Cal on a food label = 1 kcal = 4.184 kJ.
Electricity formula
$E = P \times t$. Convert P to kW (÷1000) and t to hours (÷60) before substituting. Cost = E × price per kWh.
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What you'll master
Know

Key facts

  • The hierarchy of mass units: mg, g, kg, tonne
  • The hierarchy of energy units: J, kJ, MJ
  • The calorie conversions: 1 cal = 4.184 J; 1 kcal = 4.184 kJ
  • Power–energy–time: $E \text{ (kWh)} = P \text{ (kW)} \times t \text{ (h)}$
  • 1 kWh = 3600 kJ
Understand

Concepts

  • Why electricity is billed in kWh rather than joules
  • The difference between a calorie (small) and a Calorie/kcal (food label)
  • How to convert minutes to hours before using the energy formula
Can do

Skills

  • Convert between mass units in multi-step problems
  • Convert between joules, kilojoules, and calories
  • Calculate energy consumption using $E = P \times t$
  • Calculate electricity costs from energy used and a given rate
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Key terms
Joule (J)The SI unit of energy; 1 J is the energy used when a force of 1 newton acts through 1 metre.
Kilowatt-hour (kWh)The practical unit of electrical energy; 1 kWh = energy used by a 1 kW device running for 1 hour = 3 600 000 J.
Calorie (kcal / Cal)The Calorie on food labels is actually a kilocalorie (kcal); 1 kcal = 4.184 kJ. The small calorie (cal) = 4.184 J.
Watt (W)The unit of power (rate of energy use); 1 W = 1 J per second. 1 kW = 1000 W.
Metric tonne (t)A mass unit equal to 1000 kg; not to be confused with the imperial "ton".
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The mass unit hierarchy

Mass units follow the same pattern as length: each step up the chain multiplies by 1000.

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The mass unit hierarchy
core concept

Mass units follow the same pattern as length: each step up the chain multiplies by 1000. The key is always identifying which direction you are converting.

To go smaller (e.g. kg → g): multiply by 1000. To go larger (e.g. g → kg): divide by 1000.

$1 \text{ t} = 1000 \text{ kg} = 10^6 \text{ g} = 10^9 \text{ mg}$
Real-world anchor, Medication dosages. A vitamin tablet contains 0.5 mg of vitamin B12. That sounds tiny, because it is. $0.5 \text{ mg} = 0.0005 \text{ g} = 0.0000005 \text{ kg}$. Converting in both directions reveals just how small a milligram actually is. Always identify the direction first.

Mass unit ladder: mg ÷ 1000 = g ÷ 1000 = kg ÷ 1000 = t. Each step multiplies by 1000 in the larger direction or divides by 1000 in the smaller direction. 1 tonne = 1000 kg = 1 000 000 g.

Pause, copy the mass unit ladder (mg ÷1000 g ÷1000 kg ÷1000 t), the ×1000 pattern for converting to a smaller unit, and the ÷1000 pattern for converting to a larger unit into your book.

Quick check: A truck carries 3.6 t of goods. The truck itself has a mass of 8500 kg. What is the total mass in tonnes?

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Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · MASS UNIT CONVERSION

A medication tablet contains 250 mg of active ingredient. A patient takes 3 tablets per day for 14 days. (a) How many grams of active ingredient does the patient take in total? (b) Express this as kilograms.

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Total tablets $= 3 \times 14 = 42$ tablets; Total mg $= 42 \times 250 = 10\,500$ mg
3 per day for 14 days; 42 tablets × 250 mg each.
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PROBLEM 2 · FOOD ENERGY CONVERSION

A muesli bar has an energy content of 756 kJ per serve. (a) Convert this to joules. (b) Convert this to Calories (kcal), correct to the nearest whole number.

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(a) $756 \text{ kJ} \times 1000 = \mathbf{756\,000 \text{ J}}$
kJ → J: multiply by 1000 (going to a smaller unit).
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PROBLEM 3 · ELECTRICITY COST

An electric kettle has a power rating of 2400 W. It is used for 3 minutes each morning. Electricity costs 32 cents per kWh. (a) How many kWh does the kettle use each morning? (b) What is the daily cost, to the nearest tenth of a cent? (c) What is the annual cost (365 days), to the nearest dollar?

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Convert: $P = 2400 \div 1000 = 2.4 \text{ kW}$; $t = 3 \div 60 = 0.05 \text{ h}$
Watts → kilowatts (÷1000); minutes → hours (÷60). Must convert before substituting.
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True or false: When a food label says "350 Calories", it means 350 joules of energy.

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Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Common errors · the 3 traps that cost marks

Trap 01
Forgetting to convert W to kW before using E = P × t
A 600 W fridge is 0.6 kW, not 600 kW. The formula requires P in kilowatts. If you substitute 600 directly, you get an answer 1000 times too large. Always divide watts by 1000 first.
Trap 02
Confusing Cal (kcal) with cal (calorie)
350 Calories on a food label = 350 kcal = 350 × 4.184 = 1464 kJ. Using 350 cal (small calories) would give 350 × 4.184 = 1464 J, a factor of 1000 too small. The capital C is critical.
Trap 03
Not converting minutes to hours
If an appliance runs for 45 minutes, that is 0.75 hours (45 ÷ 60). Substituting t = 45 into $E = P \times t$ gives an answer 60 times too large. Always divide minutes by 60 before substituting.

Fill the gap: A 1200 W toaster is used for 4 minutes. Converting: $P = 1200 \div 1000 =$ kW; $t = 4 \div 60 \approx$ h; $E = 1.2 \times 0.0\overline{6} \approx$ kWh.

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Revisit your thinking

Quick-fire practice · 5 calculations

Quick-fire practice · 5 calculations

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Convert: (a) 3.25 kg to g   (b) 4500 mg to g   (c) 2.4 t to kg   (d) 0.5 mg to g

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Three packages have masses of 450 g, 1.2 kg, and 800 g. Find the total mass in kilograms.

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Convert: (a) 5000 J to kJ   (b) 1050 kJ to Cal (kcal, 1 d.p.)   (c) 8700 kJ to MJ

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A pool pump is rated at 0.75 kW and runs for 8 hours per day. How many kWh does it use per day? If electricity costs 28 cents per kWh, find the weekly cost.

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A household uses 22 kWh of electricity in a day. If electricity costs 35 cents per kWh, find the daily cost in dollars.

Odd one out: Three of these are correct statements about the energy formula. Which one is wrong?

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Revisit your thinking

Earlier you estimated the kettle energy and cost. Let's calculate:

$P = 2400 \text{ W} = 2.4 \text{ kW}$; $t = 3 \text{ min} = 0.05 \text{ h}$. So $E = 2.4 \times 0.05 = 0.12 \text{ kWh}$.

Cost at 30 cents/kWh: $0.12 \times 30 = 3.6 \text{ cents}$, less than 4 cents per boil. Over a year: $3.6 \times 365 = 1314 \text{ c} \approx \$13.14$. Your kettle habit costs about $13 per year.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
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Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 33 marks

Q1. A portable speaker is charged using a 5 W USB cable for 2.5 hours. (a) Calculate the energy used to charge the speaker in kWh. (b) Convert this energy to kilojoules. (c) If electricity costs 30 cents per kWh, find the cost to charge the speaker. (3 marks)

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ApplyBand 44 marks

Q2. A household appliance runs for 3 hours at a power of 800 W. Electricity is charged at 34 cents per kWh. (a) Find the energy used in kWh. (b) Find the cost of running the appliance. (c) If the appliance is used every day, find the annual cost to the nearest dollar. (4 marks)

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AnalyseBand 54 marks

Q3. An athlete burns 2800 Cal (kcal) per day through exercise and metabolism. (a) Convert 2800 Cal to kilojoules. (b) Convert your answer to megajoules, correct to 2 decimal places. (c) If the athlete's daily food provides 12 500 kJ, by how many kilojoules does their intake compare to their energy requirement? (4 marks)

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📖 Comprehensive answers (click to reveal)

Drill 1: (a) $3250$ g (b) $4.5$ g (c) $2400$ kg (d) $0.0005$ g

Drill 2: $450\text{ g} = 0.45\text{ kg}$; total $= 0.45 + 1.2 + 0.8 = 2.45$ kg

Drill 3: (a) $5$ kJ  ·  (b) $1050 \div 4.184 \approx 250.9$ Cal  ·  (c) $8.7$ MJ

Drill 4: $E = 0.75 \times 8 = 6$ kWh/day; weekly $= 6 \times 7 = 42$ kWh; cost $= 42 \times 28 = 1176\text{ c} = \$11.76$

Drill 5: $22 \times 0.35 = \$7.70$

Q1 (3 marks): (a) $P = 0.005$ kW; $E = 0.005 \times 2.5 = 0.0125$ kWh [1]. (b) $0.0125 \times 3600 = 45$ kJ [1]. (c) $0.0125 \times 30 = 0.375$ cents [1].

Q2 (4 marks): (a) $P = 0.8$ kW; $E = 0.8 \times 3 = 2.4$ kWh [1]. (b) $2.4 \times 34 = 81.6$ cents $= \$0.816$ [1]. (c) Annual: $0.816 \times 365 = 297.84 \approx \$298$ [2].

Q3 (4 marks): (a) $2800 \times 4.184 = 11\,715.2$ kJ [1]. (b) $11\,715.2 \div 1000 = 11.72$ MJ [1]. (c) Intake $12\,500$ kJ vs requirement $11\,715.2$ kJ; intake exceeds requirement by $12\,500 - 11\,715.2 = 784.8$ kJ [2].