Science questions, answered
65 short answers from 13 guides. Each set links to the full guide for the worked examples and the reasoning behind the answer.
Density, Mass and Volume: Formula, Units and the Archimedes Test
Why does ice float on water?
Ice floats because it is less dense than liquid water, about 917 kg/m³ at 0 °C. A floating object sinks until it has displaced its own weight of liquid, so the fraction submerged equals its specific gravity divided by the liquid's. Ice therefore floats with about 92% of its volume below the surface in fresh water and about 89% in seawater.
What is Archimedes' principle?
Archimedes' principle says a submerged object is pushed up with a force equal to the weight of the water it displaces. In hydrostatic weighing, an object is weighed in air and again hanging fully under water, and the loss in mass equals the mass of displaced water, so V = (m air − m water) ÷ ρ water. A balance read to 0.01 g makes this far more precise than a measuring cylinder.
What is the density of air?
Dry air at 15 °C and 101.325 kPa has a density of 1.225 kg/m³, the sea-level value of the standard atmosphere, and at 20 °C it is 1.204 kg/m³. Gas density depends strongly on pressure and temperature: for an ideal gas, ρ = P M ÷ (R T), where M is the molar mass.
Does a litre of any liquid weigh one kilogram?
No, a litre weighs about one kilogram only for water. A litre of olive oil is about 0.915 kg, a litre of honey about 1.42 kg, and petrol at a typical 0.745 g/cm³ is lighter than water. Temperature matters as well: petrol expands by about 0.1% for every degree, so a litre bought cold contains more fuel than a litre bought warm.
What is the density of steel?
Carbon steel has a typical density of 7,850 kg/m³, or 7.85 g/cm³, so a steel part with a volume of 2 litres weighs 15.7 kg, or 34.6 lb. Aluminium is 2,700 kg/m³, and the same shape in aluminium would weigh 5.4 kg. Alloy densities are typical values, because they vary with composition.
Free Fall With and Without Air Resistance: Fall Time and Terminal Velocity
What is the terminal velocity of a skydiver?
An 80 kg skydiver falling belly to earth, modelled with a drag coefficient of 1.0 and a frontal area of 0.7 m², has a terminal velocity of about 42.8 m/s, or 154 km/h. Real belly-to-earth terminal speeds are usually quoted around 50 to 55 m/s, and a head-down skydiver can exceed 70 m/s, because posture, clothing and body size vary a lot.
Why are golf balls dimpled?
Golf balls are dimpled to trigger the drag crisis at lower speeds. A smooth sphere has a drag coefficient of about 0.47 at everyday sizes and speeds, but above a Reynolds number of about 3 × 10⁵ its drag coefficient drops sharply, to around 0.1. The dimples bring on that transition earlier, so the ball meets less drag.
Do objects fall faster at high altitude?
Yes, in air. Terminal velocity is proportional to 1 ÷ √ρ, so thinner air means a higher top speed. Air density is 1.225 kg/m³ at sea level and about 0.74 kg/m³ at 5,000 m, so a skydiver there falls about 1.29 times faster. Felix Baumgartner's 2012 jump from 39 km exceeded the speed of sound because the air was under a hundredth as dense.
How fast does a baseball hit the ground when dropped from 100 m?
A 0.145 kg baseball, 74 mm across with a drag coefficient of 0.47, has a terminal velocity of about 33.9 m/s in sea-level air. Dropped from 100 m it lands after 5.18 s at 30.7 m/s, compared with 4.52 s and 44.3 m/s in a vacuum. Drag takes a third off the impact speed and more than half off the impact energy.
When does the quadratic drag model not work?
Quadratic drag breaks down for very small or slow objects such as dust, mist and pollen, where drag is proportional to v rather than v² under Stokes' law. It also struggles with tumbling or flat objects like leaves and paper, very long falls through air of changing density, and objects such as balloons where buoyancy matters.
How Long Does an EV Take to Charge? Level 1, Level 2 and DC
How do I calculate how long an EV takes to charge?
Energy needed is usable battery capacity x (target % - start %), and time is energy needed / (charging power x efficiency). A 75 kWh battery charging from 20% to 80% needs 45 kWh. On a 7.68 kW home charger with 90% efficiency that takes 45 / (7.68 x 0.9) = 6.5 hours, or 6 hours 31 minutes.
How many kW is a 40 amp EV charger circuit?
A 40 A circuit at 240 V gives a 7.68 kW Level 2 charger. The US National Electrical Code (Article 625) treats EV charging as a continuous load, so the charger may draw only 80% of the circuit rating, 32 A. A 60 A circuit allows 48 A, or 11.52 kW, and a standard 120 V Level 1 outlet at 12 A gives 1.44 kW.
How long does DC fast charging take from 10% to 80%?
For a 75 kWh car with a 170 kW peak on a 250 kW charger, 10% to 80% takes about 23 minutes in the EV Charging Time and Cost Calculator model. Going on to 100% takes another 20 minutes for less than a third as much energy, so on a road trip two short stops are usually quicker than one long one.
Is public fast charging more expensive than charging at home?
Usually, yes. A home Level 2 charge drawing 50 kWh at 18 cents per kWh costs $9.00, or 20 cents per kWh stored. A DC session from 10% to 80% at 48 cents per kWh comes to $27.10, about 51.6 cents per kWh stored, almost three times as much. That gap is why home charging decides whether an EV saves money.
What size home EV charger do I need?
Start with your car's onboard AC limit, since there is no point paying for a 48 A circuit if the car accepts 7.2 kW. Then work out daily need: 12,000 miles a year is about 33 miles a day, and a 7.68 kW Level 2 charger adds about 25 miles an hour, so an hour and a half each night covers it.
How to Calculate pH: Strong and Weak Acids, Bases and Buffers
How do you find the pH of a strong base?
Find [OH⁻] from the base concentration, take pOH = −log₁₀[OH⁻], then pH = 14.00 − pOH at 25 °C. NaOH and KOH supply one OH⁻ each, while Ca(OH)₂ and Ba(OH)₂ supply two. So 0.01 M NaOH has pOH 2.00 and pH 12.00, and 0.01 M Ca(OH)₂ has [OH⁻] = 0.02 M and pH 12.30.
What is the pH of sulfuric acid?
A 0.01 M sulfuric acid solution has pH 1.85. The first proton of H₂SO₄ is strong but the second has pKa 1.99, so treating both as strong (pH 1.70) or only the first (pH 2.00) is wrong. Solving the second equilibrium gives [H⁺] = 0.0142 M, with only 42% of the second proton released at that concentration.
How are pKa and pH related?
pKa is −log₁₀ Ka, and the smaller the pKa, the stronger the acid. In a buffer, pH = pKa + log₁₀([A⁻] ÷ [HA]), so when the weak acid and its conjugate base are present in equal amounts the pH equals the pKa. Ten times more base raises the pH by one unit, and ten times more acid lowers it by one.
How do I make a buffer at a specific pH?
Run the Henderson-Hasselbalch equation backwards: the base-to-acid ratio [A⁻] ÷ [HA] = 10^(pH − pKa). For pH 5.00 with acetate (pKa 4.76) the ratio is 10^(0.24) = 1.74. Choose an acid whose pKa is within one unit of the target pH and keep the buffer at 10 mM or more so the equation stays reliable.
What is buffer capacity?
Buffer capacity measures how much strong acid or base a buffer absorbs per unit change in pH. For a monoprotic buffer it peaks at pH = pKa, at about 0.576 times the total buffer concentration. A 0.1 M acetic acid and 0.1 M acetate buffer has a capacity of about 115 mM per pH unit, so adding 10 mM of HCl moves its pH by under 0.1.
How to Calculate Your Carbon Footprint from Your Bills
What does CO₂e mean?
CO₂e, carbon dioxide equivalent, adds up greenhouse gases by multiplying each one by its global warming potential over 100 years. The EPA GHG Emission Factors Hub uses the IPCC Fifth Assessment Report values: 1 for carbon dioxide, 28 for methane and 265 for nitrous oxide. For fossil fuels burned at home or in a car, methane and nitrous oxide add well under 1%.
How much CO2 does a flight produce per km?
Under the UK DESNZ 2025 factors, short-haul economy flying emits 0.07435 kg CO₂e per passenger-kilometre without radiative forcing and 0.12576 kg with it, while long-haul economy is 0.06926 kg and 0.11704 kg. Business and first class are allocated more fuel because their seats take up more of the cabin. The factors already add 8% to great-circle distance.
What is the carbon footprint of a return flight from London to New York?
A return flight between London and New York, about 5,570 km each way or 11,140 passenger-km in total, is 1.30 tonnes of CO₂e in economy with radiative forcing, or 0.77 tonnes without, using the UK DESNZ 2025 factors. In business class the same return trip comes to 3.78 tonnes with radiative forcing.
How do I calculate my household carbon footprint?
Add up a year of bills and multiply each quantity by a published emission factor: kilowatt-hours by your grid's rate, therms or gallons of heating fuel, gallons of car fuel and flight passenger-kilometres, then add a diet average. A two-person US household example worked this way comes to 14.7 tonnes of CO₂e a year, or 7.4 tonnes per person.
How much CO2 does an electric car produce?
An electric car's emissions are its electricity use multiplied by the grid's emission factor. At 30 kWh per 100 miles, 11,000 miles uses 3,300 kWh, which is 1.16 tonnes of CO₂e on the US average grid and 0.66 tonnes on the California grid. A petrol car covering 11,000 miles at 28 mpg emits 3.45 tonnes from the tailpipe.
Kinetic Energy: ½mv², Speed Squared and Where the Energy Goes
Where does the kinetic energy formula ½mv² come from?
½mv² is the work done bringing a mass from rest to speed v. A net force F over distance d gives acceleration a = F ÷ m, and from rest v² = 2ad, so work W = F × d = (ma) × (v² ÷ 2a) = ½mv². Gaspard-Gustave Coriolis set out this modern form, with the factor of a half, in 1829.
How many joules are in a kilowatt-hour?
One kilowatt-hour is 3,600,000 joules. For comparison, one food kilocalorie is 4,184 J, one British thermal unit is 1,055.06 J and one foot-pound-force is 1.3558 J. A 1,500 kg car at 50 km/h carries about 145 kJ of kinetic energy, which is only about 35 kcal.
Is kinetic energy conserved in a collision?
Kinetic energy is conserved only in perfectly elastic collisions, while momentum is always conserved when no outside force acts. When a 1,500 kg car at 50 km/h hits an identical stationary car and they lock together, kinetic energy falls from 144.7 kJ to 72.3 kJ. The missing half went into heat, sound and crumpling both cars.
What is rotational kinetic energy?
Rotational kinetic energy is the energy of spin, ½Iω², where I is the moment of inertia and ω the angular speed, and it adds to the ½mv² of straight-line motion. A solid ball rolling without slipping has 40% more energy than the same ball sliding at the same speed, which is why it is slower at the bottom of a ramp.
What units should I use in the kinetic energy formula?
Use kilograms for mass and metres per second for speed, and KE = ½mv² comes out in joules with no further conversion. Divide km/h by 3.6 or multiply mph by 0.44704 to get m/s, since leaving speed in km/h inflates the answer by a factor of 13. Grams must be converted too: 8 g is 0.008 kg.
Molar Mass, Molarity, Dilutions and Half-Life: Lab Maths
How do you calculate molar mass from a chemical formula?
Add up the atomic weights of every atom in the formula. Water, H2O, is 2 × 1.008 + 15.999 = 18.015 g/mol, and glucose, C6H12O6, is 180.156 g/mol. Brackets multiply everything inside, so Al2(SO4)3 has twelve oxygens, and capital letters start symbols, so Co is cobalt while CO is carbon monoxide.
How do you calculate the molar mass of a hydrate?
Include the water of crystallisation in the formula. Copper(II) sulfate pentahydrate, CuSO4·5H2O, is 249.677 g/mol, while the anhydrous salt is 159.602 g/mol. If you weigh the blue pentahydrate crystals but use the anhydrous figure, the resulting solution ends up 36% weaker than intended.
How do you make a serial dilution?
A serial dilution applies C1V1 = C2V2 step by step. For a tenfold series from 1 M, take 1 mL of each solution into 9 mL of diluent, giving 100 mM, then 10 mM, then 1 mM and so on. Errors compound along the series, so pipette accuracy matters more than it does for a single dilution step.
What is the half-life formula?
The half-life formula is N = N₀ × (1/2)^(t / T½), or equivalently N = N₀ × e^(−λt), where the decay constant λ = ln 2 / T½. Each half-life, half of what remains decays, so 100% becomes 50%, then 25%, 12.5% and 6.25%. To find elapsed time, use t = T½ × log₂(N₀ / N).
How does carbon-14 dating work?
Carbon-14 dating uses exponential decay: a sample with a quarter of its original carbon-14 has been through two half-lives, so with the evaluated half-life of 5,700 years it is about 11,400 years old. Laboratories report conventional radiocarbon ages using Willard Libby's 5,568-year half-life, then convert them to calendar dates with calibration curves such as IntCal.
Newton's Second Law, Weight and Friction: Solving Force Problems
How do I convert newtons to pounds-force?
One pound-force (lbf) is exactly 4.4482216152605 N, so divide a force in newtons by that figure to get pounds-force. A 70 kg person weighs 686.5 N under standard gravity, which is 154.3 lbf. A 3,600 N net force on an accelerating car is about 809 lbf. One kilogram-force is 9.80665 N.
How much does a 70 kg person weigh on the Moon?
A 70 kg person weighs 113.4 N on the Moon, where g is 1.62 m/s², compared with 686.5 N on Earth. The mass stays 70 kg everywhere, because weight is mass times local gravity. The same person would weigh 259.7 N on Mars and about 1,735 N on Jupiter.
Does friction depend on contact area?
In the simple Amontons-Coulomb model of friction, no: friction force equals the coefficient of friction times the normal force, and it does not depend on the contact area. The model also predicts that kinetic friction hardly depends on sliding speed. Both predictions hold reasonably well in practice, though surface finish, moisture and temperature change the coefficients.
What is the coefficient of friction for rubber on concrete and steel on steel?
Rubber on dry concrete has a static coefficient of friction of about 1.0 and a kinetic coefficient of 0.8. Steel on steel is about 0.74 static and 0.57 kinetic, according to the textbook values in Serway and Jewett. Treat these as rough, because dirt, moisture, temperature and surface finish can move them by half or more.
How do you calculate acceleration down a slope with friction?
Once a block is sliding down a slope, its acceleration is a = g (sin θ − μk cos θ), from Newton's second law along the surface. For a block on a 25° wooden ramp with μk = 0.2, that gives 9.80665 × (sin 25° − 0.2 × cos 25°) = 2.37 m/s². The mass cancels out of the result.
Ohm's Law, Resistor Color Codes and Voltage Drop Explained
How do I read a resistor colour code?
Read a resistor colour code from the end where the bands are bunched, since gold and silver never start a code. On a four-band resistor the first two bands are digits, the third is a multiplier and the fourth is tolerance, with black 0 through white 9. Yellow, violet, red, gold is 47 × 100 = 4,700 Ω at ±5%.
What are E12, E24 and E96 resistor values?
E12, E24 and E96 are the IEC 60063 series of preferred resistor values. E12 has 12 values per decade (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82) and suits ±10% parts, E24 adds values in between for ±5%, and E96 has 96 steps per decade for ±1%. Each step is the same ratio, about 21% for E12.
What is the voltage drop formula?
For a DC or single-phase circuit, voltage drop is Vd = 2 × I × R × L, where R is resistance per unit length and L is the one-way distance, because current flows out and back. For a balanced three-phase circuit, Vd = √3 × I × R × L. For example, 100 feet of 12 AWG copper carrying 16 A at 120 V drops 6.34 V, or 5.28%.
What is the NEC 3% voltage drop rule?
The NEC 3% figure comes from informational notes in the US National Electrical Code: about 3% drop on a branch circuit or feeder and 5% in total from the service to the farthest outlet. They are advice for efficiency and performance rather than enforceable requirements in most US jurisdictions. In the UK, BS 7671 sets 3% for lighting circuits and 5% for other uses.
Is aluminium wire as good as copper?
Aluminium conducts about 61% as well as copper, so the same wire size has roughly 1.6 times the resistance and a larger voltage drop. A 250-foot, 100 A, 480 V three-phase feeder drops 1.10% in 1/0 copper but 1.81% in 1/0 aluminium. Aluminium is lighter and cheaper per ampere, so it is common for large feeders, upsized by about two AWG sizes.
Projectile Motion Equations: Range, Height, Time and the Best Angle
What are the projectile motion equations?
Projectile motion splits into independent horizontal and vertical parts. Horizontally the velocity is constant and x = v cos θ × t. Vertically gravity acts, so y = h + v sin θ × t − ½ g t², where h is the launch height and g is 9.80665 m/s². Eliminating t gives the parabola y = h + x tan θ − g x² ÷ (2 v² cos² θ).
What is the best launch angle from a height?
When a projectile is launched from above the landing level, the best angle is below 45° and satisfies tan θ = v ÷ √(v² + 2 g h). A ball launched at 10 m/s from a 20 m ledge goes farthest at 24.3°, reaching 22.62 m instead of 20.20 m for a horizontal launch. A shot put released at 13.7 m/s from 2.1 m has a drag-free optimum of 42.2°.
How far does a ball thrown horizontally off a cliff travel?
A horizontally launched projectile falls exactly like a dropped one, so the fall time is √(2h ÷ g) and the distance is that time multiplied by the launch speed. A ball rolling off a 20 m ledge at 10 m/s falls for 2.020 s and lands 20.20 m out, hitting the ground at 22.19 m/s, 63.2° below horizontal.
How do I calculate the launch angle needed to hit a target?
Use tan θ = (v² ± √(v⁴ − g(g d² + 2 y v²))) ÷ (g d), where d is the horizontal distance and y the target's height above the launch point. The ± gives a low direct shot and a high lob, and a negative value under the square root means the target is out of range. A 20 m/s throw from 1.5 m at a ground target 30 m away needs 20.2° or 67.0°.
How far does a projectile go on the Moon?
Projectile formulas scale with gravity, so on the Moon, where g is 1.62 m/s², a launch travels about six times as far and high as on Earth. A 20 m/s launch at 45° covers 246.9 m, rises to 61.7 m and stays in the air for 17.5 s, ignoring air resistance.
Speed, Distance and Download Time: Getting Rates Right
How does a current affect a boat's travel time?
Speed over the ground differs from speed through the water whenever there is a current. A boat making 12 knots through the water against a 2-knot current covers ground at 10 knots, which turns a 300 nautical mile passage from 25 hours into 30 hours. Wind has the same effect on an aircraft's speed over the ground.
How much network overhead should I allow for a download?
A 5 to 15% overhead allowance is a fair default for download estimates on a good wired connection, with more for Wi-Fi or a busy network and 0% for the theoretical best case. Protocol headers alone take around 5%. With 10% overhead, a 50 GB download on a 100 Mbit/s line takes 1 hour 14 minutes instead of 1 hour 7.
How does latency limit download speed?
A single TCP connection can only have a window of unacknowledged data in flight, so its throughput is at most the window divided by the round-trip time. With the old 64 KiB window and a 100 ms round trip, that is about 5.2 Mbit/s. Modern window scaling lifts this limit, but high latency still slows the start of every connection.
How long does a 1 TB cloud backup take to upload?
A first cloud backup of 1 TB at 20 Mbit/s upload takes over five days with modest overhead, because many cable and DSL plans upload at a fraction of their download speed. Estimates should use the upload figure from a speed test, not the headline download number, which is why providers offer to seed large backups by shipping a drive.
Why do many small files transfer slower than one large file?
Each file adds request and metadata overhead, so 10,000 small files transfer more slowly than one archive of the same total size. Compressing the files into a single archive before sending them often helps twice: there is less data to move and there are fewer round trips.
The Ideal Gas Law PV = nRT: Units, STP and When It Fails
Why must temperature be in kelvin for the ideal gas law?
The ideal gas law only works with absolute scales that start at true zero, so temperature must be in kelvin, where K = °C + 273.15. Using Celsius gives nonsense: warming a gas from 10 °C to 20 °C at fixed volume does not double the pressure, it raises it by 293.15 ÷ 283.15, about 3.5%.
Who discovered the ideal gas law?
The ideal gas law grew out of several results. Robert Boyle found in 1662 that pressure times volume is constant at fixed temperature, Jacques Charles (around 1787) and Joseph Gay-Lussac (1802) linked volume and pressure to temperature, and Avogadro proposed in 1811 that equal volumes hold equal numbers of molecules. Émile Clapeyron combined them in 1834 into PV = nRT.
How much does tyre pressure rise when the tyre gets hot?
With the volume nearly fixed, absolute pressure rises in proportion to absolute temperature. A tyre at 2.2 bar absolute at 10 °C reaches 2.2 × 313.15 ÷ 283.15 = 2.433 bar absolute at 40 °C, so the gauge reading rises from about 1.19 to 1.42 bar. That is why tyre makers specify pressures cold.
How much air is in a scuba tank?
A 12 L cylinder filled to 200 bar gauge at 15 °C holds 100.7 mol of air by the ideal gas law, which is 2.92 kg. At those pressures air has a compressibility factor Z of about 1.03 to 1.04, so the real figure is a few percent lower, roughly 2.8 kg. Diving gas-blending tables use real-gas data for that reason.
Can the ideal gas law tell how full a propane cylinder is?
No. A propane or butane cylinder contains liquid with vapour above it, and its pressure is set by the temperature, through the vapour pressure, not by how much gas is left. PV = nRT therefore cannot show how full the cylinder is; weigh it instead. Near condensation, gases like propane also deviate from the ideal law by tens of percent.
Wavelength, Frequency and Photon Energy: One Equation, Every Wave
Why is ultraviolet light harmful when visible light is not?
Each photon's energy is E = h × f, so higher-frequency light carries more energy per photon. A carbon-carbon chemical bond takes roughly 3.6 eV to break, and ultraviolet photons carry that much and more, which is why UV causes sunburn and sterilises surfaces. Visible photons fall just below it, and a 2.45 GHz microwave photon carries only about 10.1 µeV.
Why is the speed of light an exact number?
The speed of light in vacuum, c = 299,792,458 m/s, is exact because since 1983 the metre has been defined as the distance light travels in 1/299,792,458 of a second. Planck's constant and the electronvolt have also been exact since the 2019 redefinition of the SI, so conversions between wavelength, frequency and photon energy rest on fixed constants.
What are the ITU radio frequency bands?
ITU-R Recommendation V.431 numbers radio bands so that band N covers 0.3 × 10ᴺ to 3 × 10ᴺ Hz. Band 8, VHF, runs from 30 to 300 MHz and carries FM radio; band 9, UHF, runs from 300 MHz to 3 GHz and includes TV, mobile phones and 2.4 GHz Wi-Fi; band 10, SHF, covers 3 to 30 GHz, including 5 GHz Wi-Fi.
Why do wind instruments go sharp as they warm up?
The speed of sound in air rises with temperature, from 343.2 m/s at 20 °C to 349.0 m/s at 30 °C. A wind instrument's air column has a fixed length, so a faster speed of sound raises its resonant frequency and the pitch goes sharp. Concert A at 440 Hz has a wavelength of 0.780 m at 20 °C but 0.793 m at 30 °C.
How long is a quarter-wave antenna for 2.4 GHz?
A quarter-wave antenna for 2.4 GHz is about 3.1 cm long in free space, a quarter of the 12.5 cm wavelength. In practice it is trimmed a few percent shorter, because signals travel slightly slower along a conductor than in free space. The radio rule of thumb λ (m) ≈ 300 ÷ f (MHz) gives the full wavelength quickly.