Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Phase Space

A particle moves in a straight line under the action of a unknown force and its motion is observed in various situations.

The position $x$ of the particle is plotted against its velocity $v$ for the motion in the following 6 cases.

In each case, what can you say about the motion of the particle?

In which way does each travel along the curve?

Can you think of a plausible physical interpretation for each?

What can you deduce about points corresponding to equal time intervals of the motion?

Next consider the velocity-position diagrams for the following situations:

1. A rubber ball dropped perpendicular to a hard concrete floor (if you know about coefficients of restitution, you could draw this motion accurately for $e=0.7$, neglecting all friction and air resistance).

2. A large parcel dropped from a stationary and very high helicopter.

3. An air hockey puck struck hard and perpendicular to a side on an air hockey table.

Extension: Think more generally about the meaning of velocity-position diagrams and the relationship to speed-time diagrams.

## You may also like

### High Jumping

### Out in Space

Or search by topic

Age 16 to 18

Challenge Level

- Problem
- Student Solutions

A particle moves in a straight line under the action of a unknown force and its motion is observed in various situations.

The position $x$ of the particle is plotted against its velocity $v$ for the motion in the following 6 cases.

In each case, what can you say about the motion of the particle?

In which way does each travel along the curve?

Can you think of a plausible physical interpretation for each?

What can you deduce about points corresponding to equal time intervals of the motion?

Next consider the velocity-position diagrams for the following situations:

1. A rubber ball dropped perpendicular to a hard concrete floor (if you know about coefficients of restitution, you could draw this motion accurately for $e=0.7$, neglecting all friction and air resistance).

2. A large parcel dropped from a stationary and very high helicopter.

3. An air hockey puck struck hard and perpendicular to a side on an air hockey table.

Extension: Think more generally about the meaning of velocity-position diagrams and the relationship to speed-time diagrams.

How high can a high jumper jump? How can a high jumper jump higher without jumping higher? Read on...

A space craft is ten thousand kilometres from the centre of the Earth moving away at 10 km per second. At what distance will it have half that speed?