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Estimation and orders of magnitude questions
Physicists check a calculation the way you check change from a shop: a rough figure first, held ready before the exact one arrives. The rough figure lives in powers of ten, and building it takes seconds.
18 original questions · 54 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).
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.)
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.)
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.)
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.)
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.)
Explain why order-of-magnitude estimates are useful in physics.
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.
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.
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.)
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise estimation and orders of magnitude one question at a time
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