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Charging a phone battery
A battery's capacity in mAh is the charge it holds: hours to charge = capacity ÷ charger current (plus about 20% losses), and energy in Wh = capacity (Ah) × voltage
Pick a phone battery of 3000, 4000 or 5000 mAh and a 1 A, 2 A or 3 A charger. Work out how long it takes to charge from empty, how much energy it stores and how many hours the screen can stay on.
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Choosing a resistor for an LED
An LED takes a fixed forward voltage (about 2 V for red, 3.2 V for blue and white); a resistor in series takes the rest, so by Ohm's law R = (supply − LED voltage) ÷ current
Light a red, green, blue or white LED from a 5 V, 9 V or 12 V supply. Work out the resistor that lets just the right current flow, and how much power the resistor turns into heat.
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Solar panels on the roof
Energy = power × time: a 0.4 kW panel in 5 hours of sun gives about 0.4 × 5 × 0.8 = 1.6 kWh, after the losses of heat, dust and wiring
Put 2 to 10 solar panels on a roof and choose how many sunny hours a day it gets. See how many kilowatt-hours they make a day and a month, how many LED bulbs that lights for an evening, and how much CO₂ it saves against coal power.
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Comparing earthquakes
Magnitude is a logarithmic scale: each step of 1 means 10 times the shaking on a seismograph and about 32 times (10^1.5) the energy
Pick two earthquakes by their magnitude, from a small 3 to a great 8, and see how many times more the ground shakes and how many times more energy the bigger one lets out. Find why the 7.8 Gorkha earthquake of 2015 was so much worse than a 5.
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How fast do satellites go?
Gravity pulls a satellite towards the Earth and keeps it moving in a circle; in an orbit of radius r its speed is v = √(GM ÷ r), so higher satellites move slower and take longer to go round
Put a satellite in a low orbit like the International Space Station, in a GPS orbit, or in a geostationary orbit used for TV. See how far it is from the Earth's centre, how fast gravity makes it go and how long one trip round the Earth takes.
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How long does light take?
Light travels at about 299 792 km every second, so the time it takes is distance ÷ speed: a fraction of a second from a satellite, 8 minutes from the Sun and more than 4 years from the nearest star
Send light to us from a GPS satellite, the Moon, the Sun, Mars and the nearest star, and work out how long each journey takes. Find out why we see the Sun as it was eight minutes ago.
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How long is my shadow?
Light travels in straight lines, so a stick's shadow is its height ÷ tan(the sun's height): the lower the sun, the longer the shadow, and with the sun straight overhead there is no shadow at all
Stand a stick in the sun and move the sun from low in the morning sky to straight overhead. See how long the shadow is each time, and why shadows are long in the morning and evening and shortest at noon.
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If the Earth were a football
A scale model divides every size and distance by the same number: if the Earth were a 22 cm football, the Moon would be a 6 cm ball 6.6 m away and the Sun a 24 m ball 2.6 km away, so space is mostly empty
Shrink the Earth to a football, a cricket ball or a marble, and shrink everything else by the same scale. See how big the Moon and the Sun would be, and how far away they would sit.
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Fan car: how a suction fan glues a car to the road
Force = pressure difference × area: a fan that lowers the pressure under a car presses it onto the road
Switch on a fan that sucks the air out from under a sports car, and see how a small pressure difference over the car's floor gives a huge downforce, more grip and faster cornering. Then blow air in to make the car float, and fire thrusters to see when thrust beats weight and the car flies.
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A rubber-band launcher
A stretched band stores elastic potential energy E = ½kx²; on release it becomes kinetic energy ½mv², so v = √(2E ÷ m), and at 45° the pellet flies about v² ÷ g: twice the stretch gives four times the energy and twice the speed
Stretch a rubber band and fire a paper pellet. Change the band, how far you stretch it and the pellet's mass, and see the energy stored in the band, the launch speed and how far the pellet flies.
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A toy parachute
A falling object speeds up until air drag equals its weight; then it falls at a steady (terminal) speed v = √(2mg ÷ (ρCA)), so a bigger canopy falls slower
Hang a 20 g, 50 g or 100 g toy under a canopy of 0.1 to 1 m², and find how fast it falls once air drag balances its weight, and how long it takes to drop 10 m.
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Bicycle gears
Gear ratio = front teeth ÷ rear teeth: the back wheel turns that many times for each turn of the pedals, so a big ratio goes far and fast but is hard to push, and a small ratio is easy to push but slow
Change the front and rear gears of a bicycle and see how far one turn of the pedals takes you, and how fast you go at the same pedalling. Find the easy gear for a hill and the fast gear for a flat road.
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Half-life of radioactive atoms
Radioactive atoms decay at random, but in each half-life half of those left decay: after n half-lives 100% × (½)ⁿ remain, whether the half-life is hours or thousands of years
Watch a crowd of radioactive atoms decay. Choose an isotope used in hospitals or for dating old bones, let half-lives go by, and see what share of the atoms is left and how long that took.
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How far away was the lightning?
Light arrives almost at once but sound travels only about 343 m a second, so the distance to lightning in km is the seconds from flash to thunder × 0.343
Count the seconds between a lightning flash and its thunder, and work out how far away the lightning struck. Find out why a short gap means you must go indoors at once.
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How loud is safe?
Loudness is measured in decibels (dB): every 10 dB is ten times the sound energy; the level falls about 6 dB each time the distance doubles, and above 85 dB the safe listening time halves for every 3 dB more
Stand near a whisper, a conversation, busy traffic, a pressure horn or a firecracker. See the sound level in decibels, how it drops as you move away, and how long your ears can safely take it each day.
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Keeping tea hot
Hot things cool faster the hotter they are than the room: T = room + (start − room) × e^(−kt); a lid and a vacuum flask cut heat loss, giving a much smaller k
Pour tea at 90 °C into an open cup, a covered cup or a vacuum flask in a 25 °C room, and follow its temperature for an hour to see which keeps it hot and why.
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Stopping a vehicle in time
Stopping distance = thinking distance (speed × reaction time) + braking distance (v² ÷ 2μg): doubling the speed doubles the first but four-times the second
A child steps onto the road 30 m ahead. Change the vehicle's speed, the driver's reaction time and the road, and find the thinking distance, the braking distance and whether it stops in time.
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Your weight in a lift
A scale shows how hard it pushes up on you: N = m(g + a), so you seem heavier when the lift speeds up upward, lighter when it speeds up downward, and weightless in free fall
Stand on a bathroom scale in a lift. See what it reads as the lift starts going up, moves steadily, starts going down, and, if the cable snapped, falls freely.