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# What's That Graph?

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Age 14 to 18

Challenge Level

*What's That Graph? printable sheet - graphs
What's That Graph? printable sheet - processes and equations*

Here are some graphs of physical processes.

**Can you suggest plausible processes that could have given rise to each graph?**

Now click below to see eight processes that give rise to the graphs.

Can you match each process to a graph?

- A cup of tea is made and the temperature measured in degrees Celsius every second. What would the temperature - time graph look like?
- The height of the valve on a bicycle tyre above the ground is measured after each centimetre that the bicycle travels forwards. What would the height - distance graph look like?
- I throw a tennis ball straight up into the air and catch it. The height of the ball from the ground is measured over the time of the journey using freeze-frame photography. What would the height - time graph look like?
- I measure several objects using inches and then using metres, plot them on a scatter graph, and join the points. What would the metres - inches graph look like?
- I jump out of a plane and the distance fallen from the plane is measured every $0.1$ second until I open my parachute. What would the distance fallen - time graph look like whilst in freefall?
- I drive at 70 miles an hour along the motorway and note the reading on my odometer (mile counter) every 5 minutes. What would the odometer reading - time graph look like?
- I blow up a roughly spherical balloon using a balloon pump. After each pump I measure the radius of the balloon in centimetres. What would the radius - number of pumps graph look like?
- I suck water through a straw out of a large beaker at a constant rate and measure the volume of liquid remaining at various times. What would the volume-time graph look like?

**Can you suggest equations that could model each of these processes?**

Now click below to see eight equations.

Can you match each equation to the graphs and related processes?

$$y=A \sqrt[3]{x}$$ | $$y = A \sin(Bx)$$ | $$y = Ax + B$$ | $$y=\frac{A}{e^x}+B$$ |

$$y=Ax^2$$ | $$y = Ax^2 + Bx + C$$ | $$y = Ax$$ | $$y = Ax+B$$ |

**Extension**

Can you determine values for the constants A, B and C for each equation? For some, you will need to make some assumptions.

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.