Throw a small object such as a marble straight up in the air and think about the words you could use to describe its motion from the instant just after it leaves your hand to the instant just before it hits the floor.

Some of your words might involve the idea of direction.

Other words might be to do with the position of the marble, its speed or whether it is slowing down or speeding up. Underlying many of these is time.

Direction

The marble moves as it does because of the gravitational pull of the earth. We understand directional words such as up and down because we experience this pull towards the centre of the earth all the time.

The vertical direction is along the line towards or away from the center of the earth.

In mathematics a quantity which has only size, or magnitude, is called a scalar.

One which has both magnitude and a direction in space is called a vector.

Distance, position and displacement

The total distance travelled by the marble at any time does not depend on its direction. It is a scalar quantity.

Position and displacement are two vectors related to distance: they have direction as well as magnitude.

Here their direction is up or down and you decide which of these is positive.

When up is taken to be positive, down is negative.

The position of the marble is then its distance above a fixed origin, for example the distance above the place it first left your hand.

Figure 1.1

When it reaches the top, the marble might have travelled a distance of 1.25 m.

Relative to your hand its position is then 1.25 m upwards or +1.25 m.

At the instant it returns to the same level as your hand it will have travelled a total distance of 2.5 m.

Its position, however, is zero upwards.

A position is always referred to a fixed origin but a displacement can be measured from any position.

When the marble returns to the level of your hand, its displacement is zero relative to your hand but −1.25 m relative to the top.

Question 1.1.1:

(i) What are the positions of the particles A, B and C in the diagram below (Figure 1.2)?

(ii) What is the displacement of B, relative to A, and relative to C ?

Figure 1.2

(Note: Here positive direction is left to right.)

Solution 1.1.1:

(i)

The position of the particle A relative to the origin B (x=0) is -4 (OR 4 leftwards).

The position of the particle B relative to the origin B (x=0) is 0.

The position of the particle C relative to the origin B (x=0) is +5 (OR 5 rightwards).

(ii)

The displacement of B relative to A is +4.

The displacement of B relative to C is -5.

Diagrams and graphs

In mathematics, it is important to use words precisely, even though they might be used more loosely in everyday life.

In addition, a picture in the form of a diagram or graph can often be used to show the information more clearly.

Figure 1.3

The above Figure 1.3 is a diagram showing the direction of motion of the marble and relevant distances. The direction of motion is indicated by an arrow.

Below Figure 1.4 is a graph showing the position above the level of your hand against the time. Notice that it is not the path of the marble.

Figure 1.4

Question 1.1.2:

The graph in Figure 1.4 shows that the position is negative after one second (point B). What does this negative position mean?

Solution 1.1.2:

The marble is below the origin.

Note: When drawing a graph it is very important to specify your axes carefully. Graphs showing motion usually have time along the horizontal axis. Then you have to decide where the origin is and which direction is positive on the vertical axis. In this graph the origin is at hand level and upwards is positive. The time is measured from the instant the marble leaves your hand.

Notation and units:

As with most mathematics, you will see here that certain letters are commonly used to denote certain quantities. This makes things easier to follow.

Here the letters used are:

• s, h, x, y and z for position
• t for time measured from a starting instant
• u and v for velocity
• a for acceleration

The S.I. (Système International d’Unités) unit for distance is the meter (m), that for time is the second (s) and that for mass the kilogram (kg). Other units follow from this so speed is measured in meters per second, written ms-1.

Question 1.1.3:

When the origin for the motion of the marble (see Figure 1.3) is on the ground, what is its position
(i) when it leaves your hand?
(ii) at the top?

Solution 1.1.3:

(i) +1 m
(ii) +2.25 m

Question 1.1.4:

A boy throws a ball vertically upwards so that its position y m at time t is as shown in the graph below.

Figure Q1.1.4

(i) Write down the position of the ball at times t = 0, 0.4, 0.8, 1.2, 1.6 and 2.

(ii) Calculate the displacement of the ball relative to its starting position at these times.

(iii) What is the total distance travelled (a) during the first 0.8 s (b) during the 2 s of the motion?

Solution 1.1.4:

(i) 3.5 m, 6 m, 6.9 m, 6 m, 3.5 m, 0 m
(ii) 0 m, 2.5 m, 3.4 m, 2.5 m, 0 m, −3.5 m
(iii) (a) 3.4 m (b) 10.3 m

Question 1.1.5:

The position of a particle moving along a straight horizontal groove is given by where x is measured in meters and t in seconds.

(i) What is the position of the particle at times t = 0, 1, 1.5, 2, 3, 4 and 5?
(ii) Draw a diagram to show the path of the particle, marking its position at these times.
(iii) Find the displacement of the particle relative to its initial position at t = 5.
(iv) Calculate the total distance travelled during the motion.

Solution 1.1.5:

Question 1.1.6:

For each of the following situations sketch a graph of position against time.

Show clearly the origin and the positive direction.

(i) A stone is dropped from a bridge which is 40 m above a river.
(ii) A parachutist jumps from a helicopter which is hovering at 2000 m. She opens her parachute after 10 s of free fall.
(iii) A bungee jumper on the end of an elastic string jumps from a high bridge.

Solution 1.1.6:

Question 1.1.7:

The diagram is a sketch of the position–time graph for a fairground ride.
(i) Describe the motion, stating in particular what happens at O, A, B, C and D.
(ii) What type of ride is this?

Solution 1.1.7:

(i) The ride starts at t = 0. At A it changes direction and returns to pass its starting point at B continuing past to C where it changes direction again returning to its initial position at D.

(ii) An oscillating ride such as a swing boat.

Question 1.1.8:

A particle moves so that its position x meters at time t seconds is .

(i) Calculate the position of the particle at times t = 0, 1, 2, 3 and 4.
(ii) Draw a diagram showing the position of the particle at these times.
(iii) Sketch a graph of the position against time.
(iv) State the times when the particle is at the origin and describe the direction in which it is moving at those times.

Solution 1.1.8:

(Ref: Cambridge International AS and A Level Mathematics by Sophie Goldie, Series Editor: Roger Porkess, Hodder Education)