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Questions on estimation and orders of magnitude
An order-of-magnitude estimate gives a quantity to the nearest power of ten, worked from a few benchmark values and rounded arithmetic. Physicists use it to test whether a full calculation has returned a plausible size, and it quickly catches a calculator slip or a unit left unconverted.
19 original questions · 59 marks · the estimation and orders of magnitude notes · Measurements
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State what is meant by the order of magnitude of a quantity.
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The order of magnitude is the size of a quantity to the nearest power of ten (1), obtained by rounding it to the closest value of 10n (1).State the order of magnitude of (a) the mass of an apple and (b) the speed of a car on a motorway.
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(a) About 0.1 kg, so 10−1 kg (1). (b) About 30 m s−1, so 101 m s−1 (1).Give the order of magnitude of (a) 4.5 × 106 and (b) 2.0 × 10−4.
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(a) 107: orders of magnitude compare by ratio, and 4.5 is nearer 10 than 1 (the crossover is √10 ≈ 3.2) (1). (b) 10−4 (1).State the order of magnitude of (a) the diameter of an atom and (b) the height of a door.
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(a) About 10−10 m (1). (b) A door is about 2 m tall, so 100 m (2 is below the crossover of √10 ≈ 3.2) (1).A student's calculation gives the mass of a family car as 1.5 × 106 kg. Use an estimate to explain why this answer must be wrong.
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A car has a mass of order 103 kg (roughly a tonne, about the mass of fifteen adults) (1). The calculated value is about a thousand times too large, three orders of magnitude, so an error such as a slipped power of ten or a unit mix-up must have occurred (1).Estimate the number of times a human heart beats in a lifetime, and give the answer to the nearest order of magnitude. (Take about 70 beats per minute and a lifetime of about 80 years.)
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Beats ≈ 70 × 60 × 24 × 365 × 80 (1) ≈ 2.9 × 109 (1), which is of order 109 beats (1).Estimate how many 1 cm3 cubes would fit inside a 1 m3 box.
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1 m = 100 cm (1), so the number = 1003 (1) = 1 × 106 cubes, of order 106 (1).Estimate the kinetic energy of a person running, and give the order of magnitude. (Take a mass of about 70 kg and a speed of about 5 m s−1.)
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KE = ½mv2 (1) ≈ ½ × 70 × 52 (1) ≈ 875 J (1), which is of order 103 J (1).Estimate the order of magnitude of the ratio of the mass of the Earth (≈ 6 × 1024 kg) to the mass of a person (≈ 60 kg).
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Ratio ≈ (6 × 1024)/60 (1) = 1 × 1023 (1), so of order 1023 (1).Estimate the mass of the air in a school laboratory, and give the order of magnitude of your answer. (Take the density of air as 1.2 kg m−3.)
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Sensible room dimensions, e.g. 10 m × 8 m × 3 m, give a volume of about 240 m3 (1). Mass = ρV ≈ 1.2 × 240 ≈ 288 kg (1), which is of order 102 kg (1).Estimate the total volume of water a person drinks in an 80-year lifetime, in m3, and give the order of magnitude. (Take about 2 litres a day; 1 litre = 10−3 m3.)
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Volume ≈ 0.002 m3 × 365 × 80 (1) ≈ 58 m3 (1). 58 is above the crossover of √10 × 10 ≈ 32, so the order of magnitude is 102 m3 (1).Estimate the gravitational potential energy gained by a walker climbing a mountain 1000 m high, and give the order of magnitude. (Use E = mgh and take g as 10 m s−2.)
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Take the walker's mass as about 70 kg (1). E = mgh ≈ 70 × 10 × 1000 = 7 × 105 J (1). 7 is above the crossover √10 ≈ 3.2, so this rounds up to order 106 J (1).Estimate the number of breaths a person takes in one day, and hence in an 80-year lifetime, expressing each to the nearest order of magnitude. (Take about 15 breaths per minute.)
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Breaths ≈ 15 × 60 × 24 = 21600 ≈ 2 × 104 a day (1), of order 104 (1). Over 80 years: 2 × 104 × 365 × 80 (1) ≈ 6 × 108, of order 109 breaths (1).Estimate the energy needed to heat a mug of water for a hot drink, and give the order of magnitude of your answer. (Take about 0.25 kg of water, c = 4200 J kg−1 K−1 and a temperature rise of about 80°C.)
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Q = mcΔθ (1) ≈ 0.25 × 4200 × 80 (1) ≈ 84000 J (1), which is of order 105 J (1).Explain why order-of-magnitude estimates are useful in physics.
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They give a quick check on whether a detailed answer is reasonable, catching errors of many powers of ten (1). They also let you judge whether something is feasible (1), or compare the sizes of quantities, without needing precise data or a full calculation (1).A student calculates the kinetic energy of a served tennis ball to be 7.5 × 104 J. Make your own estimate of this energy, stating the values you assume, and deduce whether the student's answer is reasonable. Suggest the error the student is most likely to have made.
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Sensible benchmarks: mass of a tennis ball ≈ 0.06 kg, serve speed ≈ 50 m s−1 (1). KE = ½mv2 ≈ ½ × 0.06 × 502 ≈ 75 J, of order 102 J (1). The student's value is about 1000 times larger, three orders of magnitude, so it is not reasonable (1). A factor of 1000 in energy suggests the mass was left in grams instead of kilograms (1).A grain of sand is roughly a cube of side 0.5 mm, and an atom has a diameter of about 10−10 m. Estimate the number of atoms in the grain, and give the order of magnitude.
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Volume of grain ≈ (0.5 × 10−3)3 ≈ 1 × 10−10 m3 (1). Volume of an atom ≈ (10−10)3 = 10−30 m3 (1). Number ≈ 1 × 10−10/10−30 ≈ 1 × 1020 (1), so of order 1020 atoms (1).Estimate the number of times a car wheel turns during the car's working life, and give the order of magnitude. (Take a lifetime distance of about 150 000 km and a wheel circumference of about 2 m.)
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Distance = 150 000 km = 1.5 × 108 m (1). Turns = distance/circumference ≈ 1.5 × 108/2 ≈ 7.5 × 107 (1), which is of order 108 turns (1).The table lists the electrical appliances a household uses on a typical day, with the power of each and the time for which it is switched on. Determine the total electrical energy the household transfers in one day, give its order of magnitude in joules, and state which single appliance transfers the most energy. (1 kW h = 3.6 × 106 J.)
appliance power P / W time on each day t / h electric shower 9000 0.10 oven 2000 1.5 kettle 2500 0.20 television 90 5.0 six LED lamps, together 54 5.0 broadband router 8 24 Mark scheme
Energy of each appliance from E = Pt, for example the oven at 2000 × 1.5 = 3000 W h = 3.0 kW h (1)
The six energies are 900, 3000, 500, 450, 270 and 192 W h (1)
Total = 5312 W h ≈ 5.3 kW h (accept 5 kW h) (1)
In joules, 5.3 × 3.6 × 106 = 1.9 × 107 J, so the order of magnitude is 107 J (1)
The oven transfers the most, 3.0 kW h, although the shower draws four and a half times its power, because the oven is on for fifteen times as long (1)
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