2.1.1 Measuring distance and time

Measuring distance

1. Distance is the length of the path an object travels, and its SI unit is the metre ($m$).
2. The best instrument for measuring a distance depends on how long the distance is and on whether the path is straight or curved.
   1. A ruler marked in millimetres suits short distances of up to about $30cm$, such as the width of a textbook or the length of a toy car.
   2. A metre rule suits straight distances of up to $1m$ on a laboratory bench, such as the length of a ramp or the gap between two light gates.
   3. A tape measure suits longer straight distances, such as a $20m$ stretch of a school corridor or playground.
   4. A trundle wheel is pushed along the ground and clicks once for every metre it rolls, so it suits long distances and curved routes such as a lap of a running track.
   5. For very long journeys, such as a bus route across a town, the distance is read from a map or from the odometer of a vehicle instead of being measured directly.
3. The resolution of an instrument is the smallest change it can show, so a millimetre ruler has a resolution of $1mm$ and a tape measure marked in centimetres has a resolution of $1cm$.
4. An instrument with a finer resolution gives more precise readings, so the chosen instrument should have a resolution that is small compared with the distance being measured.
5. The start and finish points are marked clearly, for example with masking tape on a floor or chalk lines on a playground, so the same distance is used for every run.
6. The distance is measured from the zero mark of the scale, which on many rulers sits a few millimetres in from the end of the wood or plastic.
7. The ruler or tape is laid flat and straight along the path, because a tape that sags or runs at an angle gives a reading that differs from the true distance.
8. The scale is read with your eye directly in line with the mark, because looking at it from an angle shifts the reading and causes a parallax error.

Measuring time

1. A time interval is the time that passes between two events, such as a trolley crossing a start line and then a finish line, and its SI unit is the second ($s$).
2. A stopwatch suits events that last several seconds or more, such as a person walking the length of a sports hall.
3. Most stopwatches display times to $0.01s$, but this does not make a hand-timed reading accurate to $0.01s$.
4. A person needs time to see an event and then press the button, and this human reaction time delays both the start and the stop of the timing.
5. The two delays are rarely equal, so a hand-timed reading can come out either too long or too short.
6. The error matters most for short time intervals, because a fixed delay is a large fraction of a time of one or two seconds but a tiny fraction of a time of one minute.
7. Timing over a longer distance therefore reduces the effect of reaction time, because the error stays about the same size while the time being measured grows.
8. A light gate sends a narrow beam of light across a gap to a detector, and it signals an electronic timer at the instant an object breaks the beam.
   1. One light gate records how long an object takes to pass through its beam.
   2. Two light gates placed a measured distance apart record how long an object takes to travel from the first beam to the second.
9. Light gates remove human reaction time from the measurement and can resolve times of $0.001s$, so they are the best choice for short, fast events such as a trolley rolling down a ramp.
10. A data logger connected to the light gates stores every reading automatically, which removes mistakes made when reading and writing down values.
11. For an event that repeats regularly, such as the swing of a pendulum, timing many repeats and dividing by the number of repeats gives a more accurate time for one repeat.
12. For example, if $20$ swings take $36.0s$, one swing takes $\frac{36.0s}{20}=1.80s$, and the reaction-time error is shared across all $20$ swings.
13. Each timing starts and stops at the same reference point, such as the front edge of a trolley crossing a line or a pendulum bob passing the lowest point of its swing.

Choosing and improving a method

1. A good method matches the instruments to the size of the distance and the length of the time interval in each situation.
   1. For a pupil walking across a school field, a trundle wheel or tape measure marks out the distance and a stopwatch times the walk, because the time is long enough for reaction time to matter little.
   2. For a trolley rolling along a bench, a metre rule measures the distance and light gates time the motion, because the time is too short to measure reliably by hand.
   3. For a cyclist on a winding path through a park, a trundle wheel follows every bend of the path, because a tape stretched straight between the ends would miss the extra length of the curves.
   4. For a car journey between two towns, the distance comes from a map or the odometer and the time comes from a clock, because both values are large and small errors hardly affect them.
2. Every measurement is repeated at least three times, so that an anomalous result can be spotted and left out.
3. The mean of the remaining readings is found by adding them together and dividing by the number of readings, which reduces the effect of random errors.
4. Repeating cannot remove a systematic error, such as measuring every distance from the end of a ruler instead of from its zero mark, because the same error affects every reading.
5. A zero error is a common systematic error, and it is corrected by checking that an instrument reads zero before it is used.

2.1.2 Measuring speed

Calculating speed

1. Speed tells you how far an object travels in each unit of time, so a speed of $3m/s$ means the object covers $3m$ in every second.
2. Speed is calculated from a measured distance and a measured time using $speed=\frac{distance\; travelled}{time\; taken}$.
3. In symbols this is $v=\frac{s}{t}$, where $v$ is the speed, $s$ is the distance travelled and $t$ is the time taken.
4. A distance in metres divided by a time in seconds gives a speed in metres per second ($m/s$), which is the SI unit of speed.
5. The distance and the time must belong to the same part of the motion, so the time used is the time taken to cover exactly the distance that was measured.
6. Over the same distance, a shorter time means a greater speed, so a pupil who runs $100m$ in $14s$ is faster than a pupil who takes $18s$.
7. In the same time, a longer distance means a greater speed, so a cyclist who rides $150m$ in $30s$ is faster than one who rides $90m$ in $30s$.
8. Speed is found by dividing the distance by the time, and never the other way round, because time divided by distance gives the number of seconds needed for each metre.
9. A speed calculated from one distance and one time describes the whole stretch that was timed, so any speeding up or slowing down inside that stretch does not show in the answer.

Planning a speed investigation

1. Every speed measurement needs two readings, a distance measured with a suitable instrument and a time measured over exactly that distance.
2. The distance is marked out first, and the timing then starts and stops as the object passes the marks.
3. In an investigation, the independent variable is the one you change on purpose, such as the angle of a ramp.
4. The dependent variable is the one you measure to see the effect, such as the speed of a trolley.
5. The control variables are kept the same, such as the trolley, the timed distance and the release point, so that only the independent variable can change the result.
6. A longer timed distance gives a longer time, which makes the reaction-time error a smaller fraction of each reading.
7. Each speed is calculated from the mean time of repeated runs, which reduces the effect of random errors.
8. Results are recorded in a table with the independent variable in the first column and units in the column headings, not beside each number.
9. When the results are plotted, the independent variable goes on the horizontal axis and the dependent variable goes on the vertical axis.

2.1.3 Ratios, units and rates in motion

Ratio and proportion

1. A ratio compares two quantities by showing how many times bigger one is than the other, and it can be written with a colon, such as $3:1$, or as a fraction, such as $\frac{3}{1}$.
2. A ratio of two quantities measured in the same unit has no unit, because the units cancel when one is divided by the other.
3. Two quantities are directly proportional when they change by the same factor, so doubling one doubles the other and halving one halves the other.
4. At a steady speed, the distance travelled is directly proportional to the time taken, because the object covers the same distance in every second.
5. For example, a coach moving steadily along a motorway covers $24km$ in $20min$, so it covers $48km$ in $40min$ and $72km$ in $60min$.
6. Proportional reasoning means finding the scale factor between two values of one quantity and applying the same factor to the linked quantity.
7. In the coach example the time is multiplied by $3$ in going from $20min$ to $60min$, so the distance is also multiplied by $3$, from $24km$ to $72km$.
8. Two quantities are inversely proportional when multiplying one of them by a factor divides the other by the same factor.
9. For a fixed distance, the time taken is inversely proportional to the speed, so doubling the speed halves the journey time.
10. For example, a cyclist who takes $30min$ to ride to school at a steady speed would take $15min$ at twice that speed and $60min$ at half that speed.
11. A graph of two directly proportional quantities is a straight line through the origin, which is how a steady journey appears when distance is plotted against time.

Converting to SI units

1. SI units are the standard units used in science, and the SI units for motion are the metre for distance, the second for time and the metre per second for speed.
2. Many everyday measurements use other units, such as kilometres, centimetres, minutes, hours and kilometres per hour, so they are converted into SI units before they go into an equation.
3. A conversion factor is the number of smaller units in one larger unit, such as the $1000m$ in $1km$.
4. To convert from a larger unit to a smaller unit, multiply by the conversion factor, because each large unit contains many small ones.
5. To convert from a smaller unit to a larger unit, divide by the conversion factor.
6. The length conversions used in motion are $1km=1000m$, $1m=100cm$ and $1m=1000mm$.
7. The time conversions used in motion are $1min=60s$ and $1h=60min=3600s$.
   1. So $3.5km$ is $3.5\times 1000=3500m$, and $45cm$ is $\frac{45}{100}=0.45m$.
   2. Similarly, $12min$ is $12\times 60=720s$, and $2.5h$ is $2.5\times 3600=9000s$.
8. A speed has a distance unit on top and a time unit underneath, so converting a speed means converting both parts.
9. To change $km/h$ into $m/s$, multiply by $1000$ to turn kilometres into metres and divide by $3600$ to turn hours into seconds.
10. Multiplying by $1000$ and dividing by $3600$ is the same as dividing by $3.6$, so a speed in $km/h$ divided by $3.6$ gives the speed in $m/s$.
11. To change $m/s$ into $km/h$, multiply by $3.6$ instead.
12. The value in $km/h$ is always the larger number, because one kilometre per hour is a much smaller speed than one metre per second.
13. Very large or very small values can be written in standard form, so $3600s$ can be written as $3.6\times 10^{3}s$ and $0.0045m$ as $4.5\times 10^{-3}m$.

Calculating rates

1. A rate tells you how quickly one quantity changes, usually given as the change in each second, minute or hour.
2. A rate is found by dividing the change in a quantity by the time over which the change happens, $rate=\frac{change\; in\; quantity}{time\; taken}$.
3. Speed is a rate, because it is the distance travelled in each unit of time.
4. The unit of a rate is built from the units used, so metres divided by seconds gives $m/s$ and kilometres divided by hours gives $km/h$.
5. Other rates met in motion include the number of laps a runner completes per hour and the number of steps a walker takes per minute.
6. Before a rate is calculated, every quantity is converted into the units the answer needs, so a distance in kilometres and a time in minutes are changed into metres and seconds for an answer in $m/s$.
7. Two rates can only be compared when they are in the same units, so a runner at $5.0m/s$ and a cyclist at $20km/h$ are compared by converting one of them first.
8. Converting the cyclist's speed gives $\frac{20}{3.6}=5.6m/s$, so the cyclist is faster than the runner.

2.1.4 Scalars and vectors in motion

Scalar and vector quantities

1. Every physical quantity has a magnitude, which is its size given as a number with a unit, such as $40m$ or $12s$.
2. A scalar quantity has magnitude only, so a number and a unit describe it completely.
3. A vector quantity has magnitude and direction, so a number and a unit are not enough to describe it on their own.
4. Time, mass, distance and speed are scalars, because none of them involves a direction.
5. Displacement, velocity and force are vectors, because each one points or acts in a particular direction.
6. A vector can be drawn as an arrow, where the length of the arrow shows the magnitude and the way the arrow points shows the direction.
7. The direction of a vector can be given in words, such as north or down the ramp, or with a sign, where one direction is chosen as positive and the opposite direction is then negative.
8. Along a straight corridor, if movement to the right is taken as positive, then $+5m$ means $5m$ to the right and $-5m$ means $5m$ to the left.
9. A negative sign on a vector shows direction only, so it does not mean that the quantity is smaller than zero in size.

Distance and displacement

1. Distance is a scalar that measures the full length of the route travelled, including every bend and every change of direction.
2. Displacement is a vector that measures how far an object ends up from its starting point in a straight line, together with the direction.
3. Distance depends on the route taken, while displacement depends only on the start and finish positions.
4. Distance can only increase or stay the same during a journey, while displacement can increase, decrease or return to zero.
5. For a journey along a straight line with no change of direction, the distance and the magnitude of the displacement are equal.
6. For example, a pupil who walks $30m$ east along a corridor and then $10m$ back west has travelled a distance of $40m$ but has a displacement of $20m$ east.
7. A runner who completes one full lap of a $400m$ track has travelled a distance of $400m$ but has a displacement of zero, because the finish point is the start point.

Speed and velocity

1. Speed is a scalar that tells you how fast an object is moving, whatever its direction.
2. Velocity is a vector that gives the speed together with the direction of travel, such as $15m/s$ north.
3. Speed is worked out from the distance travelled, while velocity is worked out from the displacement.
4. Two cars travelling at $20m/s$ in opposite directions along the same road have the same speed but different velocities.
5. If one direction along the road is taken as positive, the velocities of the two cars are $+20m/s$ and $-20m/s$.
6. An object's velocity changes whenever its speed changes, its direction changes, or both change.
7. A car driving round a roundabout at a steady $8m/s$ has a constant speed but a changing velocity, because its direction keeps changing.
8. A tennis ball that is hit straight back at the same speed it arrived has an unchanged speed but a reversed velocity, so the sign of its velocity has changed.
9. A negative velocity means the object is moving in the negative direction, and it does not mean that the object is slowing down or moving slowly.

2.1.5 Distance-time and velocity-time graphs

Distance-time graphs

1. A distance-time graph shows the distance an object has travelled from its starting point on the vertical axis and the time on the horizontal axis.
2. The gradient of a line is its steepness, found by dividing the change in the vertical quantity by the change in the horizontal quantity.
3. On a distance-time graph the gradient is the change in distance divided by the change in time, which is the speed of the object.
4. The shape of the line therefore shows how the object is moving.
   1. A horizontal line has a gradient of zero, so the object is stationary because its distance is not changing.
   2. A straight sloping line has a constant gradient, so the object moves at a constant speed and covers equal distances in equal times.
   3. A steeper straight line shows a greater speed, because more distance is covered in each second.
   4. A curve that gets steeper shows the object speeding up, because the gradient increases as time passes.
   5. A curve that becomes less steep shows the object slowing down, because the gradient decreases as time passes.
5. When the vertical axis shows the distance from a fixed starting point, a line that slopes downwards shows the object travelling back towards the start.
6. To find the gradient of a straight line, choose two points far apart on it, read the change in distance and the change in time between them, and divide the first by the second.
7. The speed at one instant on a curved line is found by drawing a tangent to the curve at that instant and calculating the gradient of the tangent.

Velocity-time graphs

1. A velocity-time graph shows the velocity of an object on the vertical axis and the time on the horizontal axis.
2. Acceleration is the change in velocity in each second, so its unit is metres per second squared (${m/s}^2$).
3. On a velocity-time graph the gradient is the change in velocity divided by the change in time, which is the acceleration of the object.
4. The shape of the line shows how the velocity is changing.
   1. A horizontal line above the time axis shows a constant velocity, because the velocity is not changing and the acceleration is zero.
   2. A horizontal line along the time axis shows an object that is stationary, because its velocity is zero.
   3. A straight line sloping upwards shows a constant acceleration, because the velocity increases by the same amount every second.
   4. A straight line sloping downwards shows a constant deceleration, because the velocity decreases by the same amount every second and the gradient is negative.
   5. A steeper line shows a greater acceleration or deceleration.
5. A curved line shows that the acceleration itself is changing, and the acceleration at one instant is the gradient of a tangent drawn at that instant.
6. Velocity is a vector, so a line that drops below the time axis shows the object moving in the opposite direction.

Describing a journey from its graphs

1. The same journey can be drawn on both kinds of graph, and each section of one graph matches a section of the other.
   1. Walking at a steady speed gives a straight sloping line on a distance-time graph and a horizontal line above the axis on a velocity-time graph.
   2. Waiting at a bus stop gives a horizontal line on a distance-time graph and a line along the time axis on a velocity-time graph.
   3. A bus pulling away gives a curve that gets steeper on a distance-time graph and a line sloping upwards on a velocity-time graph.
   4. A bus braking to a stop gives a curve that levels off on a distance-time graph and a line sloping down to the axis on a velocity-time graph.
2. To compare two sections of a journey, compare their gradients, since the steeper section of a distance-time graph is the faster one and the steeper section of a velocity-time graph has the larger acceleration.
3. Values can also be read straight from either graph, such as the time at which an object stops or the greatest velocity it reaches.
4. The labels on the axes always decide how a graph is read, so check the quantity on the vertical axis before interpreting any line.

2.1.6 Area under a velocity-time graph

Area and distance travelled

1. The area between a velocity-time graph and the time axis represents the distance travelled by the object during that time.
2. This works because distance equals velocity multiplied by time, and on the graph the velocity is the height of the area while the time is its width.
3. The units confirm the idea, because a velocity in $m/s$ multiplied by a time in $s$ gives a distance in $m$.
4. For a constant velocity the line is horizontal, so the area underneath it is a rectangle with a height equal to the velocity and a width equal to the time.
   1. A car moving at a steady $15m/s$ for $6s$ covers $15\times 6=90m$, which is the area of the rectangle under its line.
5. For a uniform acceleration from rest the line slopes up from the origin, so the area underneath it is a triangle.
   1. The area of a triangle is $\frac{1}{2}\times base\times height$, where the base is the time and the height is the final velocity.
6. For a uniform acceleration between two velocities the area is a trapezium, which can be split into a rectangle under the starting velocity and a triangle on top of it.
   1. The trapezium can also be found in one step as $\frac{1}{2}\times (initial\; velocity+final\; velocity)\times time$.
7. A journey with several stages is split into sections, the area of each section is found, and the areas are added to give the total distance.
8. The gradient and the area give two different quantities from the same graph, because the gradient gives the acceleration while the area gives the distance.

Curved lines and counting squares

1. When the line is curved, the area under it cannot be split exactly into simple shapes, so it is estimated by counting the squares of the grid.
2. First work out the distance that one grid square represents by multiplying the velocity of its height by the time of its width.
3. Next count every whole square under the line, count each square that is half or more under the line as one square, and ignore each square that is less than half under it.
4. Multiplying the number of squares by the distance per square gives an estimate of the total distance.
5. A finer grid gives a better estimate, because less of the area is wrongly gained or lost along the edge of the curve.
6. Another method replaces the curve with a few straight lines joining points on it, which turns the area into triangles and trapezia that are quick to calculate.

Areas below the time axis

1. Velocity is a vector, so a section of a velocity-time graph below the time axis shows the object moving in the opposite direction.
2. An area above the axis represents a displacement in the positive direction, and an area below the axis represents a displacement in the negative direction.
3. The total distance travelled is found by adding the sizes of all the areas, treating every area as positive.
4. The overall displacement is found by giving areas below the axis a negative sign before adding, so these areas cancel part of the areas above the axis.
5. For example, a ball rolled up a slope that rolls back down past its starting point could have an area of $6m$ above the axis and an area of $9m$ below it.
6. The ball then travels a total distance of $6+9=15m$ but ends with a displacement of $6-9=-3m$, which places it $3m$ below its starting point on the slope.

2.1.7 Average speed for non-uniform motion

Non-uniform motion

1. An object moving at a constant speed covers equal distances in equal times, so one value of speed describes its whole journey.
2. Most real journeys are non-uniform, because the speed changes as the object speeds up, slows down and stops.
3. A bus on a town route is a typical example, because it pulls away from stops, slows at junctions, waits at traffic lights and picks up speed again on clear stretches of road.
4. The instantaneous speed is the speed at one particular moment, which is the value a car's speedometer shows.
5. For non-uniform motion the instantaneous speed changes from moment to moment, so no single reading of it describes the whole journey.
6. The average speed gives one value for the whole journey, and it equals the steady speed that would cover the same total distance in the same total time.
7. The average speed is lower than the highest instantaneous speed reached and higher than the lowest, because the fast and slow parts of the journey are combined into one value.

Calculating average speed

1. Average speed is calculated using $average\; speed=\frac{total\; distance\; travelled}{total\; time\; taken}$.
2. For a journey made in stages, the distances of all the stages are added to give the total distance.
3. The times of all the stages are added to give the total time, including every period when the object was stationary.
4. A stop is included because the journey is still taking time while the object is not moving, and leaving it out makes the average speed too large.
5. All the distances must be in the same unit and all the times must be in the same unit before they are added.
6. The average speed is not the mean of the stage speeds, unless every stage lasts for exactly the same time.
7. Over equal distances, a slow stage lasts longer than a fast stage, so the slow stage pulls the average speed down more than a simple mean of the speeds suggests.
8. On a distance-time graph, the total distance and the total time are read from the start and end points of the line, and the average speed is the gradient of a straight line joining those two points.

2.1.8 Uniform motion and uniform acceleration

Uniform motion

1. In uniform motion an object moves in a straight line at a constant speed, so it covers the same distance in every second.
2. Distance, speed and time for uniform motion are linked by $distance\; travelled=speed\times time$, an equation you need to recall.
3. In symbols this is $s=v\times t$, where $s$ is the distance in $m$, $v$ is the speed in $m/s$ and $t$ is the time in $s$.
4. Rearranging gives $v=\frac{s}{t}$ to find a speed and $t=\frac{s}{v}$ to find a time.
5. To rearrange, divide both sides of $s=v\times t$ by the quantity you want to remove, so dividing both sides by $v$ leaves $t=\frac{s}{v}$.
6. Every value is converted into SI units before it is substituted, so a time in minutes is changed into seconds and a distance in kilometres into metres.
7. For uniform motion the distance-time graph is a straight line, and its gradient is the constant speed $v$.

Uniform acceleration

1. In uniform acceleration the velocity changes by the same amount every second, so the velocity-time graph of the motion is a straight line.
2. Acceleration is linked to the change in velocity and the time by $acceleration=\frac{change\; in\; velocity}{time}$, an equation you need to recall.
3. The change in velocity is the final velocity $v$ minus the initial velocity $u$, so in symbols the equation is $a=\frac{v-u}{t}$.
4. Acceleration is measured in ${m/s}^2$, because it is a change in velocity in $m/s$ divided by a time in $s$.
5. An acceleration of $3{m/s}^2$ means that the velocity increases by $3m/s$ in every second.
6. When an object slows down, $v$ is smaller than $u$, so the acceleration comes out negative and is called a deceleration.
7. The phrase "starts from rest" tells you that $u=0m/s$, and the phrase "comes to rest" tells you that $v=0m/s$.
8. Rearranging gives $v=u+a\times t$ to find the final velocity and $t=\frac{v-u}{a}$ to find the time.
9. When the time is not known, a second equation links the two velocities, the acceleration and the distance, $v^2-u^2=2\times a\times s$.
10. You do not need to memorise $v^2-u^2=2as$, but you must be able to select it and apply it.
11. Rearranging it gives $a=\frac{v^2-u^2}{2s}$, $s=\frac{v^2-u^2}{2a}$ and $v=\sqrt{u^2+2as}$.
12. To choose an equation, list the quantities you know and the one you need, then pick the equation that contains exactly those quantities.
    1. If $u$, $v$ and $t$ are involved, use $a=\frac{v-u}{t}$.
    2. If $u$, $v$ and $s$ are involved and no time is given, use $v^2-u^2=2as$.

Measuring acceleration

1. An acceleration is found from two velocities and the time between them, so an experiment has to measure the speed at two points as well as the time taken to travel between them.
2. An interrupt card of known length is fixed upright on a trolley, so that it blocks the beam of a light gate as it passes through.
3. The speed at a light gate is the length of the card divided by the time the beam is blocked, $speed=\frac{length\; of\; card}{time\; beam\; is\; blocked}$.
4. With two light gates, the speed at each gate and the time between the gates give the acceleration from $a=\frac{v-u}{t}$.
5. A double interrupt card with two tabs gives two speeds from a single light gate, and the data logger records the time between the two tabs.
6. With one light gate and a trolley released from rest, $u=0$, so $v^2-u^2=2as$ gives the acceleration from the speed at the gate and the distance from the release point.
7. A ticker timer is an older method that prints dots on a paper tape at fixed time intervals, so dots that become further apart show a trolley speeding up.
8. Light gates are preferred because the blocking times are often less than a tenth of a second, which is far too short to measure by hand.