Food for thought

## Process/graph/equation triples

Here is each process with its graph and equation. We have repeated in blue the description from a previous section linking processes and graphs, and where appropriate have added additional comments on the equation.

• Temperature of a cup of tea over time. Graph G; $\quad y(x) = 20 + 80e^{-0.05x}$

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.

At the start, the temperature is $100^{\circ}$C, and it then decays exponentially to room temperature. The $0.05$ in the equation is a measure of how quickly it cools down.

• Height of the valve on a bicycle tyre as the bicycle moves forwards. Graph C; $\quad y(x) = 0.325\left(\sin(3.1x) + 1\right)$

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

The diameter of the wheel is assumed to be $\quantity{65}{cm}$. The $3.1$ in the equation is a number which is related to the speed of the wheel turning (and $3.1x$ is treated as an angle in radians).

• Height of a tennis ball thrown straight up and then caught. Graph H; $\quad y(x) = 10x - 4.9x^2 + 1$

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

The $4.9$ comes from half the acceleration due to gravity (measured in metres per second per second). The $10$ is the starting velocity (in metres per second), and the $1$ is the height from which the ball is thrown.

Also, once the ball reaches $1$ metre above the ground again, it is caught and remains at this height.

• Distance fallen by a parachutist jumping out of a plane. Graph D; $\quad y(x) = 4.9x^2$

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. Graph A; $\quad y(x) = 70x + 31000$

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

Here $31000$ miles is the assumed reading on the odometer at the start of the journey, and the car is assumed to be travelling at a constant $\quantity{70}{mph}$; the $x$-axis is time in hours.

• Radius of a spherical balloon as it is inflated. Graph F; $\quad y(x) = 0.6204\sqrt{x}$

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

The balloon starts with radius $0$, and is assumed to be spherical throughout the inflation process, with the volume increasing at a constant rate of $\quantity{1}{cm^3\,s^{-1}}$.

• Volume of water remaining in a cup as water is sucked out through a straw. Graph B; $\quad y(x) = 600 - 50x$

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

Here the cup starts with $\quantity{600}{ml}$ of water, and this water is removed at a constant rate of $\quantity{50}{ml\,s^{-1}}$; the whole cup is drunk in $12$ seconds.

• Distance along a tape measure measured in inches compared with distance measured in metres. Graph E; $\quad y(x) = 39.37x$

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

There are $\quantity{39.37}{inches}$ in a metre.