Master the basics - and everything after gets easier

What it is

The three equations of motion connect initial velocity, acceleration, time, distance and final velocity for steady acceleration.

Answer

Given u = 0, a = 2 m/s^2, t = 5 s. Final velocity v = u + at = 0 + 2 x 5 = 10 m/s. Distance s = ut + (1/2)at^2 = 0 + (1/2)(2)(5^2) = 25 m. So the car reaches 10 m/s and covers 25 m.

What it is

Force is what changes an object's momentum; for a fixed mass, force = mass x acceleration.

Answer

Newton's second law states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force; for constant mass this gives F = ma. Momentum is the quantity of motion of a body, equal to the product of its mass and velocity, p = mv.

What it is

Every mass attracts every other mass with a force that grows with mass and shrinks fast with distance.

Answer

The universal law of gravitation states that every body in the universe attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The force F = G m1 m2 / r^2, where G is the universal gravitational constant.

What it is

Work is force times the distance moved along it; a moving object stores that work as kinetic energy.

Answer

Work is done when a force acts on a body and the body moves in the direction of the force; work done W = F s cos(theta), where s is the displacement and theta is the angle between the force and the displacement. Kinetic energy is the energy possessed by a body due to its motion and is given by KE = (1/2) m v^2. The unit of both work and energy is the joule.

What it is

When the fastest particles escape a liquid's surface, the particles left behind are cooler.

Answer

During evaporation the fastest-moving (most energetic) particles at the surface of a liquid escape into the air. This lowers the average kinetic energy of the remaining particles, so the temperature of the liquid falls. The escaping particles also absorb latent heat from the surroundings, which is why evaporation produces a cooling effect - for example, sweating cools our body.

What it is

A mixture is a physical blend you can separate; a compound is a fixed chemical union with new properties.

Answer

In a mixture the components are just mixed physically in any ratio, keep their own properties and can be separated by physical methods like filtration. In a compound the elements are chemically combined in a fixed ratio to form a new substance with new properties, and they can be separated only by chemical methods. For example, a mixture of iron and sulphur versus the compound iron sulphide.

What it is

In any chemical reaction the total mass stays the same - atoms are only rearranged, never lost.

Answer

The law of conservation of mass states that mass can neither be created nor destroyed in a chemical reaction; the total mass of the reactants equals the total mass of the products. For example, when 12 g of carbon burns completely in 32 g of oxygen, exactly 44 g of carbon dioxide is formed (12 + 32 = 44).

What it is

Atomic number = number of protons (the element's identity); mass number = protons + neutrons.

Answer

The atomic number (Z) is the number of protons in the nucleus of an atom, which also equals the number of electrons in a neutral atom; it decides the identity of the element. The mass number (A) is the total number of protons and neutrons in the nucleus. So the number of neutrons = A - Z.

What it is

Prokaryotes (bacteria) have no true nucleus; eukaryotes (plants, animals) have a nucleus and organelles.

Answer

A prokaryotic cell has no true nucleus - its genetic material lies free in the cytoplasm - and it lacks membrane-bound organelles like mitochondria; bacteria are examples. A eukaryotic cell has a true, membrane-bound nucleus and membrane-bound organelles; plant and animal cells are examples. Eukaryotic cells are usually larger and more complex.

What it is

Meristem is a plant's growth tissue - young cells that keep dividing at tips and edges.

Answer

Meristematic tissue is made of small, thin-walled cells that keep dividing to help the plant grow. It is found at the growing regions - apical meristem at the tips of roots and shoots (which increases length) and lateral meristem or cambium (which increases thickness). Because its cells divide actively, they have dense cytoplasm and no large vacuoles.

What it is

Mixed cropping blends crops with no set pattern; intercropping plants them in definite rows.

Answer

In mixed cropping two or more crops are grown together on the same field at the same time with no fixed pattern, mainly to reduce the risk of total crop failure. In intercropping two or more crops are grown together in a definite row pattern (for example, a few rows of one crop then a few of another) to use nutrients and space efficiently and to help control pests.

What it is

The atmosphere is Earth's protective blanket of air that supplies gases and steadies temperature.

Answer

The atmosphere is a blanket of air that supports life in several ways: it provides oxygen for breathing and carbon dioxide for photosynthesis, keeps the temperature of the Earth steady by preventing sudden day-to-night changes, and shields us from the Sun's harmful ultraviolet rays through the ozone layer. Winds and the water cycle also depend on it.

What it is

Rationalising clears a surd (root) from the denominator by multiplying top and bottom by it.

Answer

Multiply the numerator and denominator by sqrt(2): (1/sqrt(2)) x (sqrt(2)/sqrt(2)) = sqrt(2) / 2. So 1/sqrt(2) = sqrt(2)/2, which is about 0.707. Rationalising removes the irrational number (surd) from the denominator.

What it is

The remainder theorem says the remainder of p(x) divided by (x - a) is simply p(a).

Answer

By the remainder theorem, the remainder when p(x) is divided by (x + 1) is p(-1). p(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1 = -1 + 3 - 3 + 1 = 0. Since the remainder is 0, (x + 1) is a factor of p(x).

What it is

A straight line meets the axes at its intercepts - put y = 0 for the x-axis, x = 0 for the y-axis.

Answer

On the x-axis, y = 0, so x + 0 = 5 gives x = 5; the line meets the x-axis at (5, 0). On the y-axis, x = 0, so 0 + y = 5 gives y = 5; it meets the y-axis at (0, 5). These two intercept points are enough to draw the straight line.

What it is

Heron's formula finds a triangle's area from just its three sides, using the semi-perimeter.

Answer

First find the semi-perimeter s = (3 + 4 + 5)/2 = 6 cm. Heron's formula gives Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[6(6 - 3)(6 - 4)(6 - 5)] = sqrt[6 x 3 x 2 x 1] = sqrt[36] = 6 square cm.

What it is

An echo is sound bouncing back off a surface and reaching you a moment later.

Answer

An echo is the sound heard again after it reflects off a hard surface and returns to the listener. To hear a distinct echo the reflected sound must reach the ear at least 0.1 second after the original, which needs the reflecting surface to be at least about 17.2 m away (since sound travels about 344 m/s in air).

What it is

The Tyndall effect is light scattering off colloid particles, revealing the beam's path.

Answer

The Tyndall effect is the scattering of a beam of light by the tiny particles of a colloid, which makes the path of the light visible. It shows that the mixture is a colloid (like milk or fog), because a true solution's particles are too small to scatter light while a colloid's particles are just the right size.

What it is

Osmosis is water moving across a membrane to balance the concentration on both sides.

Answer

Osmosis is the movement of water molecules through a selectively permeable membrane from a region of higher water concentration to a region of lower water concentration. A cell placed in pure water (a hypotonic solution) gains water by osmosis and swells; an animal cell may even burst, while a plant cell becomes firm (turgid) because of its cell wall.

What it is

A sphere's volume grows with the cube of its radius: V = (4/3) pi r^3.

Answer

Volume of a sphere = (4/3) x pi x r^3 = (4/3) x (22/7) x 7^3 = (4/3) x (22/7) x 343 = (4 x 22 x 49)/3 = 4312/3, which is about 1437.3 cubic cm.

What it is

Any object in a fluid feels an upward buoyant force equal to the weight of fluid it displaces.

Answer

Archimedes' principle states that when a body is fully or partly immersed in a fluid, it experiences an upward buoyant force (upthrust) equal to the weight of the fluid it displaces. An iron nail sinks because it displaces only a little water, whose weight is less than the nail's weight. An iron ship is hollow and wide, so it displaces a large volume of water whose weight equals the ship's weight, and so it floats.

What it is

Isotopes are atoms of the same element with the same protons but different numbers of neutrons.

Answer

Isotopes are atoms of the same element that have the same atomic number (the same number of protons) but different mass numbers (different numbers of neutrons). Because they have the same number of protons and electrons, they show the same chemical properties. For example, carbon-12 and carbon-14 are isotopes of carbon.

What it is

Xylem and phloem are the transport tissues of plants - xylem carries water, phloem carries food.

Answer

Xylem and phloem are the two conducting (vascular) tissues in plants. Xylem carries water and dissolved minerals upward from the roots to the leaves. Phloem transports the food (sugar) made in the leaves during photosynthesis to all other parts of the plant. Together they form the plant's transport system.

What it is

The mean (average) is the total of all values divided by how many values there are.

Answer

Mean = (sum of all observations) / (number of observations) = (10 + 15 + 20 + 25 + 30) / 5 = 100 / 5 = 20.

What it is

Living things are sorted into a hierarchy of groups, from Kingdom down to Species.

Answer

Classification is the arrangement of living organisms into groups and sub-groups on the basis of their similarities and differences. It makes the study of a huge variety of organisms easier and shows the evolutionary relationships between them. The main groups, from the largest to the smallest, are Kingdom, Phylum (called Division in plants), Class, Order, Family, Genus and Species. As we move down this hierarchy the number of organisms in a group decreases while their shared characteristics increase.

What it is

Density is how much mass is packed into a given volume.

Answer

Density = mass / volume = 200 / 50 = 4 g/cm^3. Since 4 g/cm^3 is greater than the density of water (1 g/cm^3), the block sinks. Packing more mass into the same volume raises the density.

What it is

Work is done when a force moves an object along its direction.

Answer

Work = force x distance = 20 x 5 = 100 joule (J). Work is done only when the object actually moves in the direction of the force; if it does not move, the work done is zero.

What it is

A moving body carries kinetic energy that depends on its mass and, strongly, on its speed.

Answer

KE = 1/2 m v^2 = 1/2 x 2 x 5^2 = 25 joule (J). Because speed is squared, doubling the speed to 10 m/s gives 1/2 x 2 x 100 = 100 J - four times as much energy.

What it is

The mole links the mass of a substance to the number of particles it contains.

Answer

Number of moles = given mass / molar mass = 36 / 18 = 2 mol. The molar mass is the mass of one mole of the substance, so dividing the given mass by it gives the number of moles.

What it is

One mole of any substance contains a fixed, huge number of particles - Avogadro's number.

Answer

Number of particles = moles x 6.022 x 10^23. So 1 mole contains 6.022 x 10^23 particles, and 2 moles contain 1.2044 x 10^24 particles. This count, 6.022 x 10^23 per mole, is Avogadro's number and is the same for every substance.

What it is

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

Answer

By the Pythagoras theorem, c^2 = a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25, so c = sqrt(25) = 5. The hypotenuse is always the longest side and lies opposite the right angle.

What it is

The net force on a body equals its mass times the acceleration it produces.

Answer

By Newton's second law, force = mass x acceleration = 10 x 5 = 50 newton (N). For a fixed force, a smaller mass gains a larger acceleration and a larger mass gains a smaller one.

What it is

Momentum is the quantity of motion in a body - the product of its mass and velocity.

Answer

Momentum = mass x velocity = 10 x 5 = 50 kg m/s. Momentum increases if either the mass or the velocity increases, and it points in the direction of motion.

What it is

Weight is the force of gravity on a body: its mass times the gravitational acceleration g.

Answer

Weight = mass x gravity = 10 x 9.8 = 98 newton (N) on Earth. On the Moon g is about 1.6 m/s^2, so the same body weighs only about 16 N, even though its mass is still 10 kg.

What it is

Mass percentage tells what fraction of a solution's mass is the dissolved solute.

Answer

Mass percentage = (mass of solute / mass of solution) x 100 = (20 / 100) x 100 = 20%. A higher solute mass, or a smaller total solution mass, gives a higher percentage.

What it is

Concentration is the amount of solute present in a given volume of solution.

Answer

Concentration = mass of solute / volume of solution = 20 / 2 = 10 g/L. Dissolving the same solute in a smaller volume gives a more concentrated (stronger) solution.

What it is

The atomic number fixes how many electrons an atom has; filling shells by the 2, 8, 8 rule gives its electronic configuration, valence electrons and valency.

Answer

Start with the atomic number Z = number of protons = number of electrons in a neutral atom. Fill the shells K, L, M, N using the 2, 8, 8, 2 pattern (the Bohr-Bury scheme) to get the electronic configuration. The electrons in the outermost shell are the valence electrons. Then: if there are 1-4 valence electrons the valency equals that number; if there are 5-7, valency = 8 - (valence electrons); and if the outer shell is complete (2 for helium, 8 for the other noble gases) the valency is 0. Example: carbon (Z = 6) is 2, 4, so 4 valence electrons and valency 4. Helium (Z = 2) has a full K shell of 2, so its valency is 0.

What it is

The mass number of an atom is the total count of protons and neutrons in its nucleus.

Answer

Mass number A = protons + neutrons = 6 + 6 = 12 (this is carbon-12). The number of protons is the atomic number and fixes which element it is; changing only the neutrons gives a different isotope of the same element.

What it is

Power is the rate of doing work - how much work is done each second.

Answer

Power = work / time = 200 / 10 = 20 watt (W). Power measures how quickly work is done, so doing the same work in a shorter time means a greater power.

What it is

Gravitational potential energy is the energy a body stores because of its height above the ground.

Answer

Potential energy = m x g x h = 2 x 9.8 x 10 = 196 joule (J). Lifting the body higher, or using a heavier body, stores more gravitational potential energy.

What it is

The speed of a wave is the product of its frequency and its wavelength.

Answer

Wave speed = frequency x wavelength = 500 x 0.5 = 250 metres per second (m/s). For a wave travelling at a fixed speed, increasing the frequency shortens the wavelength, and vice versa.

What it is

An electron dot (Lewis) structure shows only the valence electrons of an atom as dots around its symbol.

Answer

First find the number of valence electrons (the electrons in the outermost shell) from the electronic configuration. Then write the element's symbol and place that many dots around it, one for each valence electron. For example, carbon (2, 4) has 4 valence electrons, so 4 dots; oxygen (2, 6) has 6 dots; sodium (2, 8, 1) has just 1. These outer electrons are the ones involved in chemical bonding.

What it is

Acceleration is the rate at which velocity changes with time.

Answer

Acceleration = (final velocity - initial velocity) / time = (20 - 0) / 5 = 4 m/s^2. If the final velocity is less than the initial velocity the acceleration is negative, which means the body is decelerating (slowing down).

What it is

Frequency is the number of vibrations per second and is the reciprocal of the time period.

Answer

Frequency = 1 / time period = 1 / 0.5 = 2 hertz (Hz), meaning 2 vibrations per second. A shorter time period means the vibration repeats more often, giving a higher frequency.

What it is

The mean of a set of values is their sum divided by how many values there are.

Answer

Mean = sum of values / number of values = 250 / 5 = 50. The mean is the value each item would have if the total were shared out equally among them.

What it is

The Kelvin (absolute) temperature is the Celsius temperature plus 273.

Answer

Kelvin = Celsius + 273 = 25 + 273 = 298 K. The lowest possible temperature, absolute zero, is 0 K, which equals -273 C; that is why the Kelvin scale never goes negative.

What it is

The number of moles equals the number of particles divided by Avogadro's number.

Answer

Moles = number of particles / (6.022 x 10^23). So 6.022 x 10^23 particles = 1 mole, and 1.2044 x 10^24 particles = 2 moles. Avogadro's number is simply the number of particles in one mole.

What it is

The surface area of a cube is six times the area of one square face.

Answer

Surface area = 6 x side^2 = 6 x 4^2 = 6 x 16 = 96 square units. A cube has six identical square faces, so we find the area of one face and multiply by six.

What it is

In an echo or SONAR, sound travels to a surface and back, so the distance is half of speed times the total time.

Answer

The sound goes to the surface and back, covering twice the distance. So distance = (speed x time) / 2 = (340 x 2) / 2 = 340 metres. We divide by two because the measured time is for the round trip.

What it is

The mass of a sample is the number of moles multiplied by the molar mass.

Answer

Mass = moles x molar mass = 2 x 18 = 36 g. This is simply the mole formula rearranged: since moles = mass / molar mass, mass = moles x molar mass.

What it is

The time period is the time taken for one complete vibration, and it is the reciprocal of the frequency.

Answer

Time period = 1 / frequency = 1 / 2 = 0.5 second. It is the time for one full vibration, so a higher frequency (more vibrations per second) means a shorter time period.

What it is

When two bodies move directly towards each other, their relative speed is the sum of their individual speeds.

Answer

Moving towards each other, the relative speed = u + v = 20 + 10 = 30 m/s. The gap between them shrinks at 30 metres every second - faster than either car alone.

What it is

For uniformly accelerated motion, the average velocity is the mean of the initial and final velocities.

Answer

Average velocity = (u + v) / 2 = (10 + 30) / 2 = 20 m/s. This midpoint value works because, under uniform acceleration, the velocity increases at a steady rate.

What it is

The first equation of motion gives the final velocity as the initial velocity plus acceleration times time.

Answer

Final velocity v = u + a t = 5 + 2 x 5 = 5 + 10 = 15 m/s. The term a t is the extra velocity gained during the time t at constant acceleration.

What it is

To convert a Kelvin temperature to Celsius, subtract 273.

Answer

Celsius = Kelvin - 273 = 300 - 273 = 27 C. So 300 K is a comfortable 27 C. This is simply the reverse of the rule K = C + 273.

What it is

The second equation of motion gives the distance covered under uniform acceleration.

Answer

Distance s = u t + 1/2 a t^2 = 10 x 5 + 1/2 x 2 x 5^2 = 50 + 25 = 75 metres. The first term (u t) is the distance at the starting speed; the second term is the extra distance from accelerating.

What it is

The average atomic mass is the weighted mean of the masses of an element's isotopes, using their percentage abundances.

Answer

Average atomic mass = (35 x 75 + 37 x 25) / 100 = (2625 + 925) / 100 = 3550 / 100 = 35.5 u. It is a weighted average, so the more abundant isotope pulls the value closer to its own mass.

What it is

Germination percentage is the fraction of sown seeds that sprout, expressed out of a hundred.

Answer

Germination % = (seeds sprouted / seeds sown) x 100 = (40 / 50) x 100 = 80%. A higher percentage means better quality seeds and a more reliable crop.

What it is

The volume of a cone is one third of pi times the square of the radius times the height.

Answer

Volume = 1/3 x pi x r^2 x h = 1/3 x 3.14159 x 3^2 x 7 = 1/3 x 3.14159 x 9 x 7 = 65.97 cubic units. A cone holds exactly one third of the cylinder with the same base and height.

What it is

The total surface area of a cylinder is the curved side plus its two circular ends.

Answer

Total surface area = 2 pi r (r + h) = 2 x 3.14159 x 3 x (3 + 7) = 2 x 3.14159 x 3 x 10 = 188.5 square units. This is the curved surface 2 pi r h plus the two circular ends 2 pi r^2.

What it is

The surface area of a cuboid is twice the sum of the areas of its three different faces.

Answer

Surface area = 2(lb + bh + hl) = 2(3x4 + 4x5 + 5x3) = 2(12 + 20 + 15) = 2 x 47 = 94 square units. A cuboid has three pairs of identical faces, so we add the three distinct faces and double the total.

What it is

Percentage purity is the fraction of a sample that is the pure substance, expressed out of a hundred.

Answer

Percentage purity = (mass of pure substance / total mass) x 100 = (90 / 100) x 100 = 90%. The remaining 10% is impurity, so a higher percentage means a cleaner, more valuable sample.

What it is

The volume of a hemisphere is two thirds of pi times the cube of the radius.

Answer

Volume = 2/3 x pi x r^3 = 2/3 x 3.14159 x 6^3 = 2/3 x 3.14159 x 216 = 452.39 cubic units. A hemisphere is half a sphere, so its volume is half of 4/3 x pi x r^3, which is 2/3 x pi x r^3.

What it is

The slant height of a cone is found from its radius and vertical height using the Pythagoras theorem.

Answer

Slant height l = sqrt(r^2 + h^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. The radius, the vertical height and the slant height form a right triangle, so Pythagoras gives the slant.

What it is

To convert a speed from kilometres per hour to metres per second, multiply by 5 and divide by 18.

Answer

m/s = km/h x 5 / 18 = 72 x 5 / 18 = 360 / 18 = 20 m/s. The factor 5/18 comes from 1000 metres in a kilometre divided by 3600 seconds in an hour.

What it is

The third equation of motion links the final velocity to the distance travelled under uniform acceleration.

Answer

Using v^2 = u^2 + 2as = 10^2 + 2 x 2 x 100 = 100 + 400 = 500, so v = sqrt(500) = 22.36 m/s. This equation is handy because it connects velocity and distance directly, without the time.

What it is

Impulse is the product of the force and the time for which it acts, and it equals the change in momentum.

Answer

Impulse = force x time = 20 x 2 = 40 N s. Since impulse equals the change in momentum, spreading the same impulse over a longer time means a smaller force - the idea behind airbags and cushioned landings.

What it is

The wavelength of a wave is its speed divided by its frequency.

Answer

Wavelength = speed / frequency = 340 / 200 = 1.7 metres. This is the wave-speed relation v = f x lambda rearranged; at a fixed speed, a higher frequency gives a shorter wavelength.

What it is

The number of oscillations equals the frequency multiplied by the time.

Answer

Number of oscillations = frequency x time = 50 x 10 = 500. Frequency is the number of oscillations per second, so multiplying by the time gives the total count.

What it is

The number of neutrons in an atom is its mass number minus its atomic number.

Answer

Neutrons = mass number - atomic number = 12 - 6 = 6. The atomic number gives the protons, so subtracting it from the mass number (protons + neutrons) leaves the number of neutrons.

What it is

The curved surface area of a cone is pi times the radius times the slant height.

Answer

Curved surface area = pi x r x l = 3.14159 x 3 x 5 = 47.12 square units. Note that l is the slant height along the sloping side, not the vertical height of the cone.

What it is

The total surface area of a cone is its curved surface plus its circular base.

Answer

Total surface area = pi x r x (l + r) = 3.14159 x 3 x (5 + 3) = 3.14159 x 3 x 8 = 75.40 square units. This is the curved surface (pi x r x l) plus the base circle (pi x r^2).

What it is

The curved surface area of a hemisphere is two pi times the square of the radius.

Answer

Curved surface area = 2 x pi x r^2 = 2 x 3.14159 x 6^2 = 2 x 3.14159 x 36 = 226.19 square units. A full sphere has surface 4 x pi x r^2, so the curved part of a hemisphere is exactly half of that.

What it is

The diagonal of a square is its side length times the square root of two.

Answer

Diagonal = side x sqrt(2) = 10 x 1.414 = 14.14 units. This comes from the Pythagoras theorem on the two equal sides of the square: diagonal^2 = side^2 + side^2 = 2 x side^2.

What it is

To convert a speed from metres per second to kilometres per hour, multiply by 18 and divide by 5.

Answer

km/h = m/s x 18 / 5 = 20 x 18 / 5 = 360 / 5 = 72 km/h. This is the reverse of the km/h to m/s rule, which multiplies by 5/18.

What it is

The distance covered can be found from the initial and final velocities and the acceleration.

Answer

Distance s = (v^2 - u^2) / (2a) = (30^2 - 10^2) / (2 x 2) = (900 - 100) / 4 = 800 / 4 = 200 m. This is the third equation of motion rearranged to make the distance the subject.

What it is

The diagonal of a rectangle is the square root of the sum of the squares of its length and breadth.

Answer

Diagonal = sqrt(l^2 + b^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. The length, breadth and diagonal form a right triangle, so the Pythagoras theorem gives the diagonal.

What it is

The longest diagonal of a cuboid is the square root of the sum of the squares of its three dimensions.

Answer

Space diagonal = sqrt(l^2 + b^2 + h^2) = sqrt(3^2 + 4^2 + 12^2) = sqrt(9 + 16 + 144) = sqrt(169) = 13 units. It extends the Pythagoras theorem into three dimensions.

What it is

Heron's formula gives the area of a triangle from the lengths of its three sides.

Answer

First the semi-perimeter s = (a+b+c)/2 = (3+4+5)/2 = 6. Then area = sqrt(s(s-a)(s-b)(s-c)) = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units. (This is a right triangle, so 1/2 x 3 x 4 = 6 agrees.)

What it is

A chemical formula is written by criss-crossing the valencies of the cation and anion, then simplifying.

Answer

Calcium is Ca with valency 2 and chloride is Cl with valency 1. Criss-crossing, the calcium valency (2) becomes the chloride subscript and the chloride valency (1) becomes the calcium subscript, giving CaCl2. For a radical that repeats, such as (NO3), we put it in brackets: calcium nitrate is Ca(NO3)2.

What it is

The area of an equilateral triangle is root three over four times the square of its side.

Answer

Area = (sqrt(3) / 4) x side^2 = (1.732 / 4) x 6^2 = 0.433 x 36 = 15.59 square units. This is a special case of Heron's formula when all three sides are equal.

What it is

The curved surface area of a cylinder is two pi times the radius times the height.

Answer

Curved surface area = 2 x pi x r x h = 2 x 3.14159 x 3 x 7 = 131.95 square units. It is the rectangle you would get by unrolling the curved side, with width equal to the circumference and height equal to the cylinder's height.

What it is

Matter exists as solid, liquid or gas depending on how much energy its particles have and how tightly they are held.

Answer

In a solid, particles are packed in a fixed pattern and only vibrate, so it has a fixed shape and volume. Adding heat lets them break free into a liquid, where they stay close but can slide, keeping the volume but taking the container's shape. More heat spreads them far apart into a gas, which moves freely and fills all the space.

What it is

The longest diagonal of a cube is its side length times the square root of three.

Answer

Space diagonal = side x sqrt(3) = 5 x 1.732 = 8.66 units. It comes from applying Pythagoras twice - once across a face, then through the solid - giving sqrt(side^2 + side^2 + side^2) = side x sqrt(3).

What it is

The total surface area of a solid hemisphere is three pi times the square of the radius.

Answer

Total surface area = 3 x pi x r^2 = 3 x 3.14159 x 6^2 = 3 x 3.14159 x 36 = 339.29 square units. It is the curved dome (2 pi r^2) plus the flat circular base (pi r^2), which together make 3 pi r^2.

What it is

The range of a set of data is the difference between its highest and lowest values.

Answer

Range = highest value - lowest value = 80 - 20 = 60. A larger range means the data is more spread out; a small range means the values are close together.

What it is

The class mark is the midpoint of a class interval, the average of its lower and upper limits.

Answer

Class mark = (lower limit + upper limit) / 2 = (10 + 20) / 2 = 15. It is the midpoint of the interval and stands in for every value in that class when we calculate a grouped mean.

What it is

Power can also be written as force times velocity when a force moves an object at a steady speed.

Answer

Power = force x velocity = 20 x 5 = 100 watt (W). This follows from power = work / time: work = force x distance, and distance / time is the velocity, so power = force x velocity.

What it is

Each element has a one or two letter symbol, often from its English or Latin name.

Answer

Element symbols use a capital first letter and, if needed, a small second letter: Hydrogen is H, Helium He, Carbon C, Nitrogen N, Oxygen O. Some come from Latin names - Sodium is Na (natrium), Iron is Fe (ferrum) and Gold is Au (aurum). Learning them is essential because every chemical formula is written using these symbols.

What it is

A body dropped from rest gains a landing speed equal to the square root of two times g times the height.

Answer

v = sqrt(2 x g x h) = sqrt(2 x 9.8 x 20) = sqrt(392) = 19.8 m/s. This comes from v^2 = u^2 + 2as with the initial speed u = 0 and acceleration a = g. All objects gain the same speed regardless of mass (ignoring air resistance).

What it is

The time a body takes to fall from rest is the square root of two times the height divided by g.

Answer

t = sqrt(2h / g) = sqrt(2 x 20 / 9.8) = sqrt(4.08) = 2.02 s. This is the fall equation h = 1/2 g t^2 rearranged for the time. Because of the square root, quadrupling the height only doubles the time.

What it is

A body dropped from rest falls a distance equal to half g times the time squared.

Answer

s = 1/2 x g x t^2 = 1/2 x 9.8 x 3^2 = 1/2 x 9.8 x 9 = 44.1 m. Since the time is squared, the object covers much more distance in later seconds than in the first second.

What it is

An exterior angle of a triangle equals the sum of the two interior angles opposite to it.

Answer

Exterior angle = angle 1 + angle 2 = 60 + 70 = 130 degrees. This works because the exterior angle and the third interior angle sit on a straight line (180 degrees), and the three interior angles also add to 180 degrees.

What it is

Two angles are supplementary when they add up to 180 degrees.

Answer

Supplement = 180 - angle = 180 - 60 = 120 degrees. Supplementary angles add to a straight angle (180 degrees), so the angles on one side of a straight line always sum to 180.

What it is

A lever balances when the turning effect (moment = force x distance from the pivot) is equal on both sides.

Answer

A lever is balanced when the moments on both sides are equal, where a moment = force x distance from the pivot. So a small weight far from the pivot can balance a large weight close to it, as long as force1 x distance1 = force2 x distance2. For example 3 N at 4 m balances 4 N at 3 m, because both moments equal 12.

What it is

The buoyant force (upthrust) on a submerged object equals the weight of the liquid it displaces.

Answer

Upthrust = density x volume x g = 1000 x (10 / 1000) x 9.8 = 1000 x 0.01 x 9.8 = 98 N. By Archimedes' principle this equals the weight of the displaced liquid, so displacing more liquid gives a larger upthrust.

What it is

Relative density is the density of a substance compared with the density of water.

Answer

Relative density = density of substance / density of water = 2700 / 1000 = 2.7. It has no unit because it is a ratio of two densities. A value above 1 means the substance sinks in water; below 1 it floats.

What it is

The power needed to lift a load is the work done against gravity divided by the time taken.

Answer

Power = work / time = (m x g x h) / t = (50 x 9.8 x 10) / 10 = 4900 / 10 = 490 W. The work done against gravity is m x g x h; dividing by the time gives the power, so a faster lift needs more power.

What it is

The edge of a cube is the cube root of its volume.

Answer

Edge = cube root of volume = cube root of 64 = 4 units, because 4 x 4 x 4 = 64. It simply reverses the volume formula V = side^3.

What it is

The height of a triangle is twice its area divided by the base.

Answer

Height = 2 x area / base = 2 x 40 / 10 = 80 / 10 = 8 units. This is the area formula area = 1/2 x base x height rearranged to make the height the subject.