When it's time for a coffee break or a rest from the screen, one of our teasers should keep you occupied for ten minutes — or possibly longer! Deceptively simple, these problems will draw on your knowledge of physics, and also on your ability to make rough approximations based on everyday experience.
To help with the puzzles, try our 'round-number handbook' — a crib sheet with useful constants and conversions in convenient form.
The problems presented here are reproduced with permission from a series entitled "The back of the envelope", collected and edited by the late Edward M. Purcell, and published in the American Journal of Physics, a publication of the American Association of Physics Teachers.
If you fancy another conundrum, why not try the Physics Question of the Week from the University of Maryland?
problem 1:
Rubber-gloved, you are bouncing on an infinite court a basketball that is charged to 10 kV. How much energy is emitted as electromagnetic radiation, per bounce?
problem 2:
If a teacup could be made impervious to nucleons, how many neutrons could it hold, at absolute zero, without running over? How many protons?
problem 3:
How big an asteroid could you escape from by jumping?
problem 4:
Can a helium balloon lift the tank that the helium came in?
problem 5:
At room temperature in still air, how long can a pencil remain balanced on its point? How long at absolute zero?
problem 6:
Estimate the length of the longest free path experienced by any nitrogen molecule in the lowest kilometre of the Earth's atmosphere within the past billion years.
problem 7:
What is the largest number of airplanes that could be in blindly random flight over the continental United States, between 5,000 and 10,000 metres altitude, without risking more than one fatality per billion passenger-kilometres as a result of mid-air collisions?
problem 8:
If the iron in the Earth's core were made into a wire as long as the radius of the visible Universe, what would be the diameter of the wire?
problem 9:
The 'Planck mass' is defined as sqrt ( h c / G). As energy, what is this worth in litres of gasoline?
problem 10:
A star like the Sun is just about visible to the naked eye if it is 50 light years away. If you want to put up an Earth satellite in the form of a reflecting sphere that can be seen as it passes over at night, what is the smallest diameter it may have?
problem 11:
A paper cup on the table is empty, except for the air it contains. The surrounding air is perfectly still. What is the best guess for the time it will take diffusion to replace half the air molecules in the cup with new molecules: a second, a minute, an hour or a day?
problem 12:
Define a 'Sun day' as the amount of energy received by the whole Earth from the Sun in one day. The world's coal reserves have been estimated at 10 Sun days. How many cubic kilometres of coal does that amount to? How does it compare with the amount of carbon in the Earth's atmosphere, of which about 1 molecule in 3,000 is carbon dioxide?
problem 13:
If the energy stored in the Earth's magnetic field could be drained with no losses and used to supply the world's demand for electrical power, how long would it last?
problem 14:
A guitar string tuned to G (392 Hz) sags in the middle, owing to its own weight, by what distance?
problem 15:
If the electrons in one raindrop could be removed from the Earth without removing the protons, by how much would the potential of the Earth be increased?
problem 16:
A copper wire 1 km long is connected across a 6-volt battery. How long does it take a conduction electron to drift around the circuit, at room temperature?
problem 17:
A perfect heat engine, or combination of heat engines, has as its only input equal amounts of hot water at 90 °C and cold water at 10 °C. As its only output it squirts all this water out in one high-speed jet. What can you say about the temperature and speed of the water in the jet?
problem 18:
How far from a 1-kilowatt radio transmitter is the r.m.s. electric field strength equal to that in the cosmic microwave background radiation?
problem 19:
If water is dispersed as a fog of droplets all of the same diameter, and if that diameter can be chosen to achieve the greatest opacity for a given amount of water, estimate the amount of water, in grams per cubic metre, required to reduce the range of visibility through the fog to about 10 metres.
problem 20:
The baseball player Monty Carlo is a genuine .250 hitter; in other words, for every official time at bat, the probability that he gets a hit is 0.250. In Carlo's record for the season, which included 300 times at bat, what is likely to be the length of his longest slump — that is, run of consecutive hitless times at bat? What is the probability that his actual batting average for the season exceeded .300?
problem 21:
A loudspeaker aperture 30 cm in diameter is putting out 5 W of acoustic power at a frequency of ~500 Hz. Estimate the amplitude of vibration of the air at the aperture.
problem 22:
On the beach at noon, with a clear sky, you took a good photograph at f/8 and 1/500 s. At midnight the same scene is lit by the full Moon. You want to make an equally well exposed picture. You have a tripod to hold the camera for a time exposure, and the aperture can be opened to f/2.8. What would be a good exposure time to try?
problem 23:
If energy equal to the world's annual electrical energy output could be used for hoisting rocks, how big a mountain could be built?
problem 24:
Before 1998, the best limit on the mass of the photon came from measurements of Jupiter's magnetic field by the Pioneer 10 spacecraft. Leverett Davis, Jr et al. (Phys. Rev. Lett. 35, 1402-1405; 1975) used these measurements to show that the range of the electromagnetic force must be at least 5 x 1010 cm. How large is the implied upper bound on the rest mass of the photon? If a green photon races an X-ray photon across the visible Universe, how far behind will it finish?
problem 25:
A basketball dropped from the top of a tall building lands on the street below. How high will it bounce?
problem 26:
The box containing a 50-watt light bulb promises an output of 900 lumens and a life of 750 hours. Is this bulb destined to emit as much as one mole of visible photons during its life?
problem 27:
When you put cream in your coffee, which causes the larger increase in entropy: the mixing of cream and water, or the heat exchange between cream and water?
problem 28:
In order of magnitude, the energy stored in ocean waves is as much as the Earth receives from the Sun in what length of time?
problem 29:
Would an electron exposed only to solar radiation pressure and gravity be expelled from the Solar System?
problem 30:
What is the probability that a straight line drawn from the Earth in an arbitrary direction (but not towards the Sun) will hit a star in our Galaxy? Assume 1011 solar-type stars and 10 kpc to the Galactic Centre.
problem 31:
A _____ of water contains about as many molecules as there are _____s of water in all of the oceans. What goes in the blank: drop, teaspoon, tablespoon, cup, quart, gallon, barrel or tonne?
problem 32:
Could a snowflake cooled to 10 µK be lifted with an ordinary permanent magnet acting on the induced nuclear polarization?
problem 33:
Is it likely or unlikely that your next breath will contain an atom of nitrogen that was in your first breath?
problem 34:
If the Moon is receding from the Earth at 4 centimetres per year, how much power is being dissipated in tidal friction?
problem 35:
How fast can a 10-mg water drop spin without flying apart? (Order of magnitude estimate only; aerodynamic forces to be ignored.)
problem 36:
If the energy transmitted by a 100-kilovolt power line 50 kilometres long is used to refine aluminium, about how long will it take to produce an amount of aluminium equal to that in the cables? An hour, a day, a week, a month, or a year?
problem 37:
How large, in order of magnitude, is the deflection of a light ray that grazes a neutron star? A light ray that grazes a galaxy?
problem 38:
Electromagnetic radiation inside your eyeball consists of two components: (a) 310 K blackbody radiation and (b) visible photons that have entered through the pupil. In order of magnitude, what is the ratio of the total energy in the second form to that in the first when you have your eyes open in a well lit room?
problem 39:
A ribbon 2-cm wide is drawn at a constant speed of 10 cm s-1 into a vat of oil and then vertically out of it. The density of the oil is 0.8 g cm-3 and its viscosity is 0.015 N m-2. Estimate the rate, in cm3 s-1, at which the oil is being carried away by the ribbon.
problem 40:
Would all the paper ever manufactured suffice to cover the land areas of the Earth?
problem 41:
A 60-W bulb lit for a year takes how many barrels of oil?
problem 42:
Rayleigh scattering would limit how far we could see horizontally even if the Earth were flat. Estimate the limit, given that liquid nitrogen has a refractive index of 1.2 for visible light and a density of 0.8 g cm-3.
problem 43:
The ratio of the effect of the Moon on ocean tides to the effect of the Sun is approximately 7/3. Using that fact and anything you have observed with your eyes, find the ratio of the mean density of the Moon to that of the Sun.
problem 44:
If a person when breathing were perfectly efficient at extracting oxygen from the air, how much air would supply the person's daily requirement of oxygen?
problem 45:
Imagine a galaxy which contains 1011 stars like the Sun, more or less evenly distributed within a sphere of radius 50,000 light years. This galaxy collides head on with a similar galaxy toward which it had been moving at a speed of 300 km s-1. How many stellar collisions are to be expected?
problem 46:
In the near-perfect interstellar vacuum, a proton has captured an electron in a circular orbit of 1-micrometre radius. How soon will this atom emit an ultraviolet photon?
problem 47:
Given that the temperature in the Earth's crust increases at the rate of 20° C per kilometre of depth, how cold would the Earth become with the Sun turned off?
problem 48:
What fraction of the Moon's path about the Sun is convex towards the Sun?
problem 49:
If all the energy released by a burning candle were emitted as 5500-Å photons, how many candle years of illumination would a 150-gramme candle provide?
problem 50:
A hydrogen molecular ion, H2+, accelerated to 1 MeV, passes through a metal foil 100-Å thick. Its only electron is thereby stripped off without significantly slowing the nuclei. By roughly how large an angle will the subsequent trajectories of the two protons diverge?
problem 51:
Is 'g' at the surface of a gold nucleus greater or less than 980 cm s-2?
problem 52:
How long will it take a ½-in-diameter steel ball bearing to sink one mile in the ocean?
problem 53:
If M is the magnitude of an earthquake on the Richter scale, the total strain energy released, in joules, is believed to be related to M as follows1: log10E = 2.4 × M – 1.2. What is the magnitude of a 'one-megaton' earthquake? Describe an event of Richter magnitude M = 1.
1. Press, F. & Siever, R. Earth 2nd edn 411 (Freeman, San Francisco, 1978).
problem 54:
About what fraction of the atoms of 1-carat diamond are on the surface? (One carat is defined as 0.2 g.)
problem 55:
Compare the energy released by completely burning a lump of coal with the kinetic energy of a similar lump of coal in low Earth orbit.
problem 56:
A typical interstellar gas cloud my contain 20 hydrogen atoms per cm3, with a velocity distribution appropriate to a temperature of 100 K. How often does a particular H atom collide with another H atom?
problem 57
A helicopter that weighs 2 tonnes fully loaded has a single rotor, the tips of which sweep out a circle 15 m in diameter. Calculate the minimum engine power required for hovering.
problem 58:
Starting from diagonally opposite corners of an empty chessboard, the white and black kings execute random walks. They move alternately and blindly, each step being randomly chosen as one of the eight, the five, or the three legal moves from the king's current position. The game ends when one king, on its Nth move, captures the other. Predict within a factor of 2 the mean N over a large number of games.
problem 59:
If it takes 3 hours to cook a 5 kg turkey, how long should it take to cook a 10 kg turkey?
