SQA National 5 Physics

Two kinds of quantity

A scalar quantity has a size (magnitude) only. A vector quantity has a size and a direction. Saying that a runner has travelled 400 m tells you how far, and saying she has moved 400 m due north tells you where she has ended up relative to where she started. The first is a scalar statement and the second a vector one.

Distance is the total length of the path taken. Displacement is the straight-line distance from the start to the finish, in a stated direction. A lap of a 400 m track gives a distance of 400 m but a displacement of zero, because the runner finishes where she began. Speed is how fast something is moving along its path, whereas velocity is speed in a stated direction, so a car going round a roundabout at a steady 10 m/s has a constant speed and a changing velocity.

Direction is given as a compass point, as a bearing (an angle clockwise from north, written with three figures such as 053), or as an angle to a stated line. A vector pointing one way is given a positive sign and one pointing the opposite way a negative sign when the motion is in a straight line.

Adding vectors: the resultant

The resultant of two or more vectors is the single vector that has the same effect as all of them together. Vectors are added tip to tail.

In one dimension

When the vectors point along the same line, add those in the same direction and subtract those in opposite directions. A boat with a velocity of 4.0 m/s through the water in a river flowing at 1.5 m/s in the same direction has a resultant velocity of $4.0 + 1.5 = 5.5$ m/s. Going against the flow, the resultant is $4.0 - 1.5 = 2.5$ m/s in the direction the boat is pointing.

At right angles

When two vectors are at 90 degrees to each other, draw them tip to tail; the resultant is the third side of a right-angled triangle, from the start of the first to the end of the second. There are two ways to find it.

By calculation, the size comes from Pythagoras' theorem and the angle from trigonometry:

By scale diagram, choose a scale (1 cm for 10 N, say), draw the first vector, draw the second from its tip at a right angle, join the start to the finish, then measure the length and convert it by the scale and measure the angle with a protractor.

Worked example

A walker goes 6.0 km north and then 8.0 km east in 2.0 hours.

The distance is $6.0 + 8.0 = 14$ km.

The displacement is the resultant:

The angle from north follows from $\tan\theta = 8.0 / 6.0$, which gives $\theta = 53^\circ$, so the displacement is 10 km on a bearing of 053.

The average speed is $14 / 2.0 = 7.0$ km/h and the average velocity is $10 / 2.0 = 5.0$ km/h on a bearing of 053.

Speed, velocity and time

Average speed is the distance travelled divided by the time taken, and average velocity is the displacement divided by the time taken. As equations:

$d$ is distance in metres (m), $s$ is displacement in metres (m) in a stated direction, $\bar{v}$ is the average speed or average velocity in metres per second (m/s), and $t$ is time in seconds (s). The equation is rearranged to $\bar{v} = d/t$ or $t = d/\bar{v}$ as needed.

A cyclist who travels 450 m at an average speed of 7.5 m/s takes

When the motion is not steady, the average tells you nothing about any one instant. A car that travels 30 km in 0.5 hours has an average speed of 60 km/h even though it will have been faster and slower at different points. The instantaneous speed is the speed at one particular moment, which is what a speedometer shows.

Measuring average speed

The aim is to find the average speed of a trolley rolling down a slope.

1. Draw a start line and a finish line on the slope and measure the distance between them with a metre stick or tape.
2. Release the trolley from rest at the start line, start a stopwatch as it crosses the start line and stop it as it crosses the finish line.
3. Calculate $\bar{v} = d/t$.
4. Repeat several times and take the mean of the times, which reduces the effect of random errors.

Using two light gates connected to an electronic timer, one at each line, removes the human reaction time from the measurement. Reaction time, typically a fraction of a second, is the largest source of uncertainty with a stopwatch, and it matters most when the total time is short. Measuring the distance from the wrong part of the trolley each time, or reading the tape from an angle (parallax error), adds a smaller uncertainty.

Measuring instantaneous speed

An instantaneous speed is found by measuring the speed over a very short stretch of the motion.

1. Attach a card of known length to the top of the trolley.
2. Place a single light gate, connected to a timer, so that the card passes through the beam. The timer records how long the beam is blocked.
3. Release the trolley and record the time.
4. Calculate the speed as the length of the card divided by the time the beam was blocked.

A card 0.050 m long that blocks the beam for 0.020 s gives

This is the average speed over the length of the card. Because the card is short and the time is so small, it is very close to the speed at the light gate. A shorter card gives a better approximation, while a card that is too short gives a time that is difficult to measure accurately.