Weekly Challenge 49: Get in line

7415 /www/nrich/html/content/id/7415/ 1 2011-06-20T16:17:39 <mdoxml xmlns:ns0="http://nrich.maths.org/mdo" version="1.0"> <ul id="buttonBar"> <li> <a href="http://nrich.maths.org/4935&part=">Try this next</a> </li> <li> <a href="http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means">Read all about it</a> </li> <li> <a href="https://nrich.maths.org/discus/messages/27/27.html">Ask NRICH</a> </li> <li> <a href="http://nrich.maths.org/5966&part=">Warm-up problem</a> </li> <li> <a href="http://nrich.maths.org/7411">Last week's challenge</a> </li> </ul> <div> </div> 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 <ol> <li>A cup of tea is made and the temperature measured in degrees Celsius every second. What would the temperature-time graph look like?</li> <li>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?</li> <li>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?</li> <li>I measure several objects using inches and then using metres. What would the inches-metres graph look like? </li> <li>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?</li> <li>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? </li> <li>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?</li> <li>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?</li> <li>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?</li> </ol> <div> Line Graphs </div> <div> </div> <table border="1" style=""> <tbody> <tr cellpadding="0"> <td cellpadding="0" style="" width="200"> <mdo:image alt="" height="150" src="Chart%201.png" width="230" /> </td> <td cellpadding="0" style="" width="200"> <mdo:image alt="" height="150" src="Chart%202.png" width="230" /> </td> <td cellpadding="0" style="" width="200"> <mdo:image alt="" height="150" src="Chart%203.png" width="230" /> </td> </tr> <tr cellpadding="0"> <td> <mdo:image alt="" height="150" src="Chart%204.png" width="230" /> </td> <td> <mdo:image alt="" height="150" src="Chart%205.png" width="230" /> </td> <td> <mdo:image alt="" height="150" src="Chart%206.png" width="230" /> </td> </tr> <tr cellpadding="0"> <td> <mdo:image alt="" height="150" src="Chart%207.png" width="230" /> </td> <td> <mdo:image alt="" height="150" src="Chart%208.png" width="230" /> </td> <td> <mdo:image alt="" height="150" src="Chart%209.png" width="230" /> </td> </tr> </tbody> </table> <div> </div> Equations <div> </div> <div>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.</div> <div> </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 1. A cup of tea is made and the temperature measured in degrees Celsius every second. What would the temperature-time graph look like? $$y=\frac{A}{2^t}+B$$ y = temp, t = time <mdo:image height="206" width="309" src="Chart%202.png" alt="b"></mdo:image> 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? $$y = A \sin(Bt)$$ <mdo:image height="206" width="309" src="Chart%204.png" alt="d"></mdo:image> y = angle, t = time 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? <mdo:image height="206" width="309" src="Chart%208.png" alt="h"></mdo:image> $$y=Ax^2$$ y = height, x = time 4. I measure several objects using inches and then using metres. What would the inches-metres graph look like? <mdo:image height="206" width="309" src="Chart%203.png" alt="c"></mdo:image> $$y = Ax$$ y = inches, x = metres 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 freefall? <mdo:image height="206" width="309" src="Chart%201.png" alt="a"></mdo:image> $$y = A x(x-B) + C$$ y = distance, x = time 6. I drive at 70 miles an hour along the motorway and note the reading on my mileometer every 5 minutes. What would the plot of mileometer v minutes passes graph look like? <mdo:image height="206" width="309" src="Chart%209.png" alt="i"></mdo:image> $$y = Ax+B$$ y = milometer, x = minutes 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? <mdo:image height="206" width="309" src="Chart%207.png" alt="g"></mdo:image> $$ y=\frac{A}{x}$$ y = distance, x = force 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? <mdo:image height="206" width="309" src="Chart%205.png" alt="e"></mdo:image> $$y=A r^3$$ y = no. of pumps, r = radius 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? $$y = Ax + B$$ y = volume, x = time <mdo:image height="206" width="309" src="Chart%206.png" alt="f"></mdo:image> </mdoxml> 2 3 0 0 0 1 1 Weekly Challenge 49: Get in line Match the graphs, the processes and the equations. sfh10 Steve - Development