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

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

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

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 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

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

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

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 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

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

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

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

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

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

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

Distance is the total path length covered; displacement is the shortest straight-line distance from start to finish, with direction.

Answer

Distance is the actual length of the path travelled by a body, irrespective of direction; it is a scalar quantity and is always positive. Displacement is the shortest distance measured from the initial to the final position of the body in a particular direction; it is a vector quantity and can be positive, negative or zero. Distance is always greater than or equal to the magnitude of displacement.

What it is

Speed is the rate of change of distance; velocity is the rate of change of displacement, so it has direction.

Answer

Speed is the distance travelled by a body per unit time; it is a scalar quantity. Velocity is the displacement of a body per unit time in a given direction; it is a vector quantity. The SI unit of both speed and velocity is the metre per second (m/s).

What it is

Acceleration is the rate of change of velocity with time.

Answer

Acceleration a = (final velocity - initial velocity) / time = (v - u)/t = (30 - 10)/5 = 20/5 = 4 m/s^2. So the car accelerates at 4 metres per second squared.

What it is

For uniform acceleration, the three equations of motion relate u, v, a, t and s.

Answer

The three equations of motion for uniform acceleration are: v = u + at; s = ut + (1/2)a t^2; and v^2 = u^2 + 2as. Here u is the initial velocity, v is the final velocity, a is the acceleration, t is the time and s is the displacement. They are valid only when the acceleration is uniform (constant).

What it is

The equations of motion solve problems where a body moves with uniform acceleration.

Answer

Given u = 0 (starts from rest), a = 2 m/s^2, t = 5 s. Using s = ut + (1/2)a t^2 = 0 x 5 + (1/2)(2)(5^2) = 0 + (1/2)(2)(25) = 25 m. So the body covers 25 metres.

What it is

A body moving in a circle at constant speed is accelerating because its direction keeps changing.

Answer

In uniform circular motion, a body moves along a circular path with a constant speed. However, its direction of motion changes continuously at every point. Since velocity depends on both speed and direction, the velocity is changing even though the speed is constant. A changing velocity means the body is accelerating, so uniform circular motion is an accelerated motion.

What it is

On a velocity-time graph, the slope gives acceleration and the area under the graph gives displacement.

Answer

On a velocity-time graph, the slope of the line represents the acceleration of the body, since acceleration is the change in velocity per unit time. The area enclosed between the graph line and the time axis represents the distance (displacement) travelled by the body. A straight, slanting line indicates uniform acceleration, while a horizontal line indicates uniform velocity.

What it is

A body stays at rest or in uniform motion unless an external unbalanced force acts on it; this tendency is inertia.

Answer

Newton's first law of motion states that a body at rest will remain at rest, and a body in uniform motion will continue to move in a straight line with the same speed, unless it is acted upon by an external unbalanced force. Inertia is the natural tendency of a body to resist any change in its state of rest or of uniform motion. The mass of a body is a measure of its inertia.

What it is

Inertia is of three kinds - inertia of rest, inertia of motion and inertia of direction.

Answer

Inertia of rest is the tendency of a body to remain at rest, for example dust falls off a carpet when it is beaten because the dust tends to stay at rest while the carpet moves. Inertia of motion is the tendency of a moving body to keep moving, for example a passenger jerks forward when a moving bus stops suddenly. Inertia of direction is the tendency of a body to keep moving in the same direction, for example mud flies off tangentially from the wheel of a moving vehicle.

What it is

The rate of change of momentum of a body is proportional to the applied force, giving F = ma.

Answer

Newton's second law states that the rate of change of momentum of a body is directly proportional to the applied unbalanced force and takes place in the direction of the force. This gives the equation F = ma. For a 5 kg body with acceleration 2 m/s^2, F = ma = 5 x 2 = 10 N. So a force of 10 newtons is required.

What it is

Momentum is the product of the mass and velocity of a body; it is a vector.

Answer

Momentum is the quantity of motion possessed by a body, equal to the product of its mass and velocity; it is a vector quantity in the direction of velocity. Momentum p = mv = 2 x 5 = 10 kg m/s. So the ball has a momentum of 10 kilogram metre per second.

What it is

For every action there is an equal and opposite reaction, acting on two different bodies.

Answer

Newton's third law of motion states that for every action there is an equal and opposite reaction. The action and reaction forces are equal in magnitude and opposite in direction, but they act on two different bodies, so they do not cancel each other. For example, when a swimmer pushes the water backwards (action), the water pushes the swimmer forward (reaction).

What it is

In the absence of an external force, the total momentum of a system stays constant.

Answer

By conservation of momentum, total momentum before firing = total momentum after firing. Before firing both are at rest, so total momentum = 0. After firing: (mass of bullet x velocity of bullet) + (mass of gun x recoil velocity) = 0. So 0.02 x 200 + 4 x V = 0, giving 4 + 4V = 0, so V = -1 m/s. The gun recoils at 1 m/s in the opposite direction.

What it is

Every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

Answer

Newton's 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 formula is F = G m1 m2 / r^2, where G is the universal gravitational constant, m1 and m2 are the masses and r is the distance between them.

What it is

Mass is the amount of matter in a body and stays constant; weight is the gravitational force on it and changes with g.

Answer

Mass is the quantity of matter contained in a body; it is a scalar, measured in kilograms, and remains constant everywhere. Weight is the force with which a body is attracted towards the centre of a planet; it is a vector given by W = mg, measured in newtons, and changes from place to place. On the Moon the acceleration due to gravity is about one-sixth of that on Earth, so an object's weight on the Moon is one-sixth of its weight on Earth, although its mass stays the same.

What it is

When a body falls only under gravity, it is in free fall with acceleration g (about 9.8 m/s^2).

Answer

In free fall the stone starts from rest, so u = 0, and its acceleration is g = 9.8 m/s^2. Using v = u + gt = 0 + 9.8 x 3 = 29.4 m/s. So the velocity after 3 seconds is 29.4 metres per second, directed downward.

What it is

Thrust is the force acting perpendicular to a surface; pressure is thrust per unit area.

Answer

Thrust is the force acting on a body perpendicular to its surface; its SI unit is the newton (N). Pressure is the thrust acting per unit area of the surface, given by pressure = thrust / area. The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m^2.

What it is

A body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces.

Answer

Buoyancy is the upward force exerted by a fluid on a body immersed in it, also called upthrust. Archimedes' principle states that when a body is immersed fully or partially in a fluid, it experiences an upward buoyant force that is equal to the weight of the fluid displaced by the body. This is why objects appear to lose weight in a liquid.

What it is

Relative density is the ratio of the density of a substance to the density of water; it decides floating or sinking.

Answer

Relative density = density of the substance / density of water = 800 / 1000 = 0.8. Since the relative density (0.8) is less than 1, the substance is less dense than water, so it will float on water. Relative density has no unit because it is a ratio of two densities.

What it is

Work is done when a force moves a body in the direction of the force; W = F s.

Answer

Work done W = force x displacement in the direction of the force = F x s = 20 x 5 = 100 J. So the work done is 100 joules. If the force is perpendicular to the displacement, the work done is zero.

What it is

Kinetic energy is the energy possessed by a body due to its motion, equal to (1/2)mv^2.

Answer

Kinetic energy KE = (1/2) m v^2 = (1/2) x 2 x 10^2 = (1/2) x 2 x 100 = 100 J. So the kinetic energy is 100 joules. Because KE depends on v^2, doubling the speed makes the kinetic energy four times larger.

What it is

Gravitational potential energy is the energy possessed by a body due to its height, equal to mgh.

Answer

Gravitational potential energy PE = m g h = 5 x 10 x 4 = 200 J. So the body has a potential energy of 200 joules. This energy can be converted into kinetic energy if the body falls.

What it is

Energy can neither be created nor destroyed; it only changes from one form to another, and the total stays constant.

Answer

The law of conservation of energy states that energy can neither be created nor destroyed; it can only be transformed from one form to another, and the total energy of an isolated system remains constant. For a freely falling body, at the top it has maximum potential energy and zero kinetic energy; as it falls, potential energy decreases and kinetic energy increases, but at every point the sum of potential and kinetic energy (the total mechanical energy) remains the same.

What it is

Power is the rate of doing work; its commercial unit of energy is the kilowatt-hour (kWh).

Answer

Power is the rate of doing work, or the work done per unit time: power = work / time. Here power = 6000 / 20 = 300 W. So the power is 300 watts. The commercial unit of electrical energy is the kilowatt-hour (kWh), where 1 kWh equals 3.6 x 10^6 joules.

What it is

An echo is a sound heard again after reflecting from a distant surface; it needs the reflector to be far enough for a 0.1 s gap.

Answer

In time 2 s the sound travels to the cliff and back, so the total distance travelled = speed x time = 340 x 2 = 680 m. This is twice the distance to the cliff, so the distance to the cliff = 680 / 2 = 340 m. To hear a distinct echo, the reflecting surface must be at least about 17 m away (so the time gap is at least 0.1 s).