Food for thought

Process/graph pairs

Here is each graph matched to its process.

• Temperature of a cup of tea over time.

Newton’s law of cooling says that the rate of cooling is proportional to the difference between the temperature of the object and the ambient temperature. Here the initial temperature is assumed to be $100^{\circ}$C, and the room temperature is $20^{\circ}$C.

• Height of the valve on a bicycle tyre as the bicycle moves forwards.

Measuring from the ground, the height of the valve oscillates between $0$ and the diameter of the wheel.

• Height of a tennis ball thrown straight up and then caught.

Assuming constant acceleration due to gravity and neglecting factors such as air resistance leads to a parabola.

• Distance fallen by a parachutist jumping out of a plane.

Near the start of the jump, long before the parachutist opens the parachute, air resistance is negligible and there is just the constant acceleration due to gravity to consider.

• Reading on the odometer (mile counter) of a car driving on a motorway.

The speed of the car is fairly constant, so the distance travelled is a straight line graph.

• Radius of a spherical balloon as it is inflated.

The volume increases at a constant rate, so the radius increases like the cube root.

• Volume of water remaining in a cup as water is sucked out through a straw.

The water is removed at a constant rate, so the volume decreases linearly.

• Distance along a tape measure measured in inches compared with distance measured in metres.

There is a conversion factor between inches and metres, which leads to a straight line graph.