Forces and motion, Year 11: distance, speed and velocity

Displacement

HintIt is the vector partner of distance.

The whyDistance is simply how far something has travelled along its path, with no direction: a scalar. This quantity ignores the route and looks only at the start and the finish, and it always comes with a direction, such as '300 m due north'.

Distance 400 m; displacement zero

HintOne of the two only cares where she began and where she ended.

The whyDistance counts every metre of the path, so one lap is 400 m. Displacement is the straight line from start to finish, and those are the same place, so it is zero. This is the clearest case of the two quantities giving different answers for one journey.

50 m north

HintThe second leg undoes part of the first.

The whyAlong a straight line, movements in opposite directions subtract: 80 m north followed by 30 m south leaves you 80 − 30 metres north of the start. The distance walked is different: 80 + 30 = 110 m. A displacement needs its direction as well as its size.

direction

HintIt is the extra piece of information that 'due north' adds to '5 km'.

The whyA scalar has size only. Saying you have walked 5 km tells nobody where you are now; to know that, they need the displacement, which carries the missing piece. The same split separates speed from velocity.

The distance — the displacement is the straight line from start to finish, which is shorter

HintOne follows the route; the other cuts the corner.

The whyDistance follows the route, so it is 3 + 4 = 7 km. Displacement is the single straight line joining the start to the finish; a scale drawing shows it is 5 km long, pointing 53° north of east. The size of a displacement can equal the distance travelled, but can never be more than it.

720 m

HintCheck the unit of the time before you multiply.

The whyDistance travelled = speed × time, or s = v t, with s in metres, v in metres per second and t in seconds. Two minutes is 120 s, so s = 6 × 120. AQA expects this equation to be recalled.

180

HintPut the distance into metres, then rearrange so that time stands alone.

The whyStarting from s = v t, the time is t = s ÷ v. The distance must match the speed's unit: 4.5 km = 4500 m, so t = 4500 ÷ 25. That is three minutes.

1.5; 3; 6

HintEach one is double the one before it.

The whyThese are the values AQA expects pupils to recall. Real speeds vary with age, fitness, the ground and how far the person has to go, so they are typical figures, not exact ones. They are useful for checking whether a calculated answer is sensible.

330

HintIt is a little over three hundred.

The whySound travels roughly a third of a kilometre every second in air. The exact figure changes with the temperature of the air, which is why some books quote 340 m/s; AQA's typical value is the one to use in its exams unless a question gives another.

About 30 m/s

HintA mile is roughly 1600 m and an hour is 3600 s.

The why70 miles is about 112 000 m, and dividing by the 3600 seconds in an hour gives about 31 m/s. A handy rule is that a speed in mph is a little more than twice the same speed in m/s. Knowing rough values like this for cars, trains and aeroplanes is part of the specification.

fitness; terrain

HintThink about who the person is, and about the route in front of them.

The whyA fit adult on a flat road moves faster than a tired child on a steep, muddy path, and nobody keeps up a sprint for ten kilometres. This is why the walking, running and cycling speeds are given as typical values.

The car's speed kept changing during the journey

HintThink of junctions, traffic lights and open road.

The whyHardly anything moves at one unchanging speed: a car slows for bends and stops at lights, then goes faster on clear road. Dividing the total distance by the total time gives the average, which hides all of those changes.

Velocity

HintTwo cars can show the same reading on the dial yet differ in this, if one heads north and the other south.

The whySpeed says only how fast; this quantity says how fast and which way — '20 m/s due east'. It is a vector. Change either the speed or the direction and it has changed, which is why a car turning a corner at a steady speed is still changing it.

No — equal speeds, but opposite directions

HintCheck what the two dials read, then check where each bonnet is pointing.

The whyVelocity is speed together with direction, so both parts must match before two velocities are the same. These cars would be written as 30 mph east and 30 mph west — the same size, pointing opposite ways.

accelerating; velocity; speed

HintThree different words: one for motion that is changing, one that includes which way, one that does not.

The whyOne journey needs all three words. How fast is one idea, how fast and which way is a second, and how quickly either of those is changing is a third. Everyday speech blurs them; physics asks for the right one on purpose.

distance

HintIt is what a car's mileometer adds up.

The whyEach vector describing motion has a scalar twin with the direction left out: velocity and speed, displacement and its partner. The scalar tells you how much; the vector tells you how much and which way.

The bus is changing direction

HintA speedometer reports only one of the two things that make up velocity.

The whyAcceleration is any change in velocity, and velocity includes which way you are heading. Moving in a circle, the speed can stay the same while the heading turns all the time, so the velocity is constantly changing — which is why you feel pushed sideways in your seat even though the dial is steady.