7033
/www/nrich/html/content/id/7033/
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2011-02-01T00:00:01
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<ul id="stemLinks">
<li><a href="http://nrich.maths.org/6044">Warm-up</a></li>
<li><a href="http://nrich.maths.org/7424">Try this next</a></li>
<li><a href="http://nrich.maths.org/5933">Think higher</a></li>
<li><a href="http://nrich.maths.org/6631">Read: mathematics</a></li>
<li><a href="http://plus.maths.org/content/how-does-gravity-work">Read: science</a></li>
<li><a href="http://plus.maths.org/content/illusory-universe">Explore further</a></li>
</ul>
<div> </div>
<br></br>
<p>In the problem Reaction Timer, two methods for testing the speed of reactions were introduced.</p>
<div style="float: right;"><mdo:image alt="dropping ruler experiment" height="219" src="reactions.jpg" width="217"></mdo:image></div>
<br></br>
<p>The second experiment suggested getting a friend to drop a ruler and measuring how far along it you could manage to catch. The shorter the distance, the quicker your reactions.<br></br>
<br></br>
<span style="font-weight: bold;">If two people caught the ruler at $15$cm and $30$cm do you think the first person's reactions are twice as fast as the second person's?</span><br></br>
<br></br>
<span style="font-style: italic;">To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling. We will assume that:</span></p>
<ul>
<li><span style="font-style: italic;">the ruler can be modelled as a single particle, rather than something long and thin,</span></li>
<li><span style="font-style: italic;">there is no air resistance on the ruler as it falls.</span></li>
</ul>
<div>Objects falling under gravity accelerate at a rate of approximately $9.8$ metres per second per second (usually written as $9.8ms^{-2}$) - that is, each second their velocity increases by $9.8ms^{-1}$.</div>
<br></br>
<div>For an object falling from rest, we can draw a table showing the velocity at regular intervals:</div>
<br></br>
<table style="" border="1">
<tbody>
<tr>
<td style="text-align: left;">Time (seconds)</td>
<td style="text-align: center;">0</td>
<td style="text-align: center;">1</td>
<td style="text-align: center;">2</td>
<td style="text-align: center;">3</td>
<td style="text-align: center;">4</td>
</tr>
<tr>
<td style="text-align: center;">Velocity (metres per second)</td>
<td style="text-align: center;">0</td>
<td style="text-align: center;">9.8</td>
<td style="text-align: center;">19.6</td>
<td style="text-align: center;">29.4</td>
<td style="text-align: center;">39.2</td>
</tr>
</tbody>
</table>
<br></br>
<p>You could represent this graphically - what would the graph look like?<br></br>
<br></br>
We can work out the distance travelled at each point in time, by multiplying the average velocity by the time taken. Complete the table below:<br></br>
</p>
<table style="" border="1">
<tbody>
<tr>
<td>Time (seconds)</td>
<td>0</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
</tr>
<tr>
<td>Distance travelled (metres)</td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br></br>
<p>You could represent this graphically - what would the graph look like?<br></br>
<br></br>
Using your graph, or by deriving a formula, can you confirm your answer to the question posed at the start of the problem?<br></br>
<br></br>
The average person's reaction time is about $0.2$ seconds. Where would you expect them to catch a ruler, according to your formula or graph?<br></br>
<br></br>
Try the experiment for yourself and see if your reaction times are quicker or slower than average.<br></br>
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<span class="editorial">Herschel of the European School of Varese
sent us a very succinct solution.</span><br></br>
<br></br>
Speed-time graph:<br></br>
<br></br>
Since gravity exerts a constant force, the acceleration is also
constant: $9.8ms^{-2}$. Therefore, speed is proportional to time
and the speed-time graph shows a straight line (with a gradient of
$9.8$).<br></br>
<br></br>
<br></br>
Distance-time graph:<br></br>
<br></br>
The distance travelled depends on the speed, which increases over
time. Therefore, the distance travelled in the first second is less
than that travelled during the second, which is in turn less than
that travelled during the third. The graph is therefore a
half-parabola with the gradient increasing over time. This means
that it takes longer for the ruler to travel the first $15cm$
compared to the next $15 cm$, and a reading of $30 cm$ therefore
indicates a reaction time that is slower, but not twice as slow!<br></br>
<br></br>
To look at it algebraically, distance is calculated by the formula
$d=\frac{1}{2}\times a\times t^2$. This means that the distance is
NOT proportional to the reaction time, but instead to the SQUARE of
the reaction time. Double the distance therefore means a reaction
time that is 1.41 (i.e. the square-root of $2$) times slower rather
than twice as slow. Rearranging the formula above to make $t$ the
subject gives us $t=\sqrt{\frac{2d}{a}}$, which means the reaction
time for $15cm$ is $\sqrt{0.03}=0.17$ seconds, while that for
$30cm$ is $\sqrt{0.06}=0.24$ seconds - only $0.07$ seconds slower,
or $41$%.<br></br>
<br></br>
As for the "typical" $0.2$ seconds reaction time, the typical
distance should be $0.5\times 9.8\times0.2^2=0.196m$ which is
$19.6cm$. I found it hard to get consistant results for my own
reactions - my distances ranged from $15cm$ ($0.17$ seconds) to
$25cm$ ($0.23$ seconds). Incidentally, it's funny to note that a
distance of $0.204$ meters ($20.4$ centimeters) corresponds
precisely to a time of $0.204$ seconds! <br></br>
<br></br>
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<h3>Why do this problem?</h3>
<a href="http://nrich.maths.org/7033">This problem</a> brings together several mathematical ideas: modelling, using quadratic graphs to solve problems, rearranging quadratic formulas, and interpreting results in the context of a real-world problem.<br></br>
<h3>Possible approach</h3>
<div>This problem follows on from <a href="http://nrich.maths.org/6044&part=">Reaction Timer</a>. Perhaps spend some time working on the second experiment before working on this problem.</div>
<br></br>
<div>Begin the lesson with this question on the board:</div>
<br></br>
<div style="font-weight: bold; text-align: left; margin-left: 40px;">If two people caught the ruler at 15 and 30cm do you think the first person's reactions are twice as fast as the second person's?</div>
<br></br>
<div>Give learners a chance first to think about it on their own, and then to agree an answer with their partner, before discussing it with the rest of the class.</div>
<br></br>
<div>Then pose the question:</div>
<br></br>
<div style="margin-left: 40px; font-weight: bold;">What is the relationship between the distance travelled by the ruler and the reaction time?</div>
<br></br>
<div>Allow a short time for learners to speculate about what sort of relationship they might expect to discover. Then hand out this worksheet (<a href="/content/id/7033/ruler.doc">Word</a>, <a href="/content/id/7033/ruler.pdf">pdf</a>) for learners to work on the problem in pairs.</div>
<div>If necessary, bring the class together to talk about units of acceleration and velocity and explain the notation $ms^{-1}$ and $ms^{-2}$, and to clarify any other new ideas met in this task.</div>
<br></br>
<div>Towards the end, give the class time to carry out the experiment and see how their reaction times compare with the average.</div>
<h3 style="text-align: left;">Key questions</h3>
<div>As an object falls, when is it travelling slowest/fastest?</div>
<div>What does the graph of distance against time tell you about the change in distance when the time doubles?</div>
<div>Do you think the modelling assumptions are reasonable?</div>
<h3>Possible extension</h3>
<div>The Stage 5 problem <a href="http://nrich.maths.org/5987&part=">Cannon Balls</a> explores vertical motion of a cannon ball and solving equations of vertical motion.</div>
<br></br>
When considering the validity of the modelling assumptions, learners could sketch how they think the graph might be changed if air resistance was taken into account.<br></br>
<h3>Possible support</h3>
<div>Work together as a class filling in the first few values of the time/distance table.</div>
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<mdoxml version="1.0">Gravity causes the ruler to accelerate as it falls - the longer it
has been falling, the faster it will be travelling!<br></br></mdoxml>
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To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...
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