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.