### Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

### Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

### Cushion Ball

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?

# Weekly Challenge 49: Get in Line

##### Stage: 4 and 5 Challenge Level:

When measurements are made in science it often makes sense to plot the resulting pairs of values on a chart and join the points with a smooth curve or a line. This allows us to make predictions of measurements in future experiments or at values other than those that we have measured. Often we can also make a mathematical model in which we suggest that the process will be described by an mathematical equation. Sometimes the equation comes entirely from theory; often an experiment will be needed to work our the numerical values of the constants in these equations.

In this problem, $9$ physical situations, their proposed equations and graphs have been mixed up and shown below.

Which can you match up? What is the interpretation of the variables $x$ and $y$ in each case for the equations? What units would be needed to label the axes in a reasonably accurate way?

Can you identify the physical interpretation of three key points on each of the graphs?

Processes
1. A cup of tea is made and the temperature measured in degrees Celsius every second. What would the temperature-time graph look like?
2. The pendulum of a grandfather clock swings to and fro and the angle of the bob from the vertical is measured every 100th of a second. What would the angle-time graphs look like?
3. 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?
4. I measure several objects using inches and then using metres. What would the inches-metres graph look like?
5. 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-time graph look like whilst in free fall?
6. I drive at 70 miles an hour along the motorway and note the reading on my milometer every 5 minutes. What would the plot of milometer v minutes passes graph look like?
7. I bring two magnets of the same polarity together directly in a line in a sequence of steps. Starting from 1 metre, I halve the remaining distance each time and measure the force felt between the magnets. What would the force-distance graph look like?
8. 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 number-of-pumps vs radius graph look like?
9. 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?
Line Graphs

Equations

In each of these equations, $A$, $B$ and $C$ are constants which you might be able to determine theoretically, or which might need an experiment to determine. In each case $x$ and $y$ are the variables which are being measured.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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