A-level Mathematics Resources

Going in circles: Making sense of A-level Maths: Mechanics 2

If you feel like your Mechanics 2 lessons are going in circles, or you don’t think you’ve got the potential to get work done, this post may be helpful to you.  Remember, with great power comes great energy transfer per second.

Before I get too carried away, let me take a moment and pick up a sofa:

Sofa Moments

Moments are a great way to start the course.  Straight away you’re diving into a concept they thought they had sussed (equilibrium) and saying “yeah, but if I put the two equal and opposite forces here and here, the book spins round – does that look like equilibrium to you?”  We have to extend our definition to cope with rigid bodies rather than particles, but it gives scope for solving so many more problems.  The sofa one I like in particular because it explains the “I’ve got the heavy end” phenomena in a way that M1 – modelling everything as particles – would miss completely.  The most useful thing is that, like resolving forces in any direction you choose, you can take moments about any point and generate more information to solve a problem.  Next time you’re building a cabin out of railway sleepers, take a moment and write a worksheet.  Some of you may not have got round to building a cabin yet, so feel free to share a moment with me:

Sleeper Moments

(This idyllic cabin beside the river where I grew up in rural Leicestershire is actually available for holidays #shamelessplug)

Centres of mass naturally follows on from moments, and even if the class won’t be studying volumes of revolution any time soon, it can’t hurt to throw a bit of quick calculus their way here, even if it’s just the hand-wavy kind.  For instance, the centre of mass of a triangular lamina lies on each median line (from each corner to the midpoint of the opposite line).  Imagine cutting the triangle into many small rectangles.  The centre of mass of each is on the median, therefore so will their combined centre of mass.

triangle

I brought in a small cardboard box with a rock duct-taped to one end of the inside, and we threw it around a bit to get a feel for non-uniform objects.  Remember, the centre of mass of any object can be found experimentally by suspending it from a point, balancing it on a point or even spinning it as you throw it (what this last one lacks in accuracy it makes up for in student engagement).

Energy is the first really big challenge of the course, and it should not be underestimated how confusing students can find energy questions.  My booklet, All About Energy, makes for a nice introduction.  Taking inspiration from BetterExplained’s ADEPT method of introducing a concept, I’ve filled the booklet with analogies, pictures and some real facts and figures (let’s face it – how many of us can find the elastic potential energy in a bungee rope but don’t know how much energy is in a bowl of Weetabix?)  Best served as a full-colour, A5 booklet, it doesn’t go into great detail with the nuts and bolts, but makes a good accompaniment to your introduction:

All About Energy

Getting to grips with the distinctions between force, energy and power is crucial.  But it’s great to be able to apply what we know about energy immediately to problems which it would be possible to solve with SUVAT equations, but by no means straight-forward:

SUVAT vs Energy

My next example is usually a roller-coaster drawn on the board, with loop-the-loops, strange twists and turns, and definitely not constant acceleration.  By knowing the height at any point (and neglecting – for now – resistance forces) the speed can be calculated easily.  This will be handy later, when applying energy to circular motion.  Add in friction (keep it constant for now), and the only extra information you need is the length of the track.

Energy is definitely one of those topics where I’ll need all colours of whiteboard pen available.  I describe kinetic, gravitational potential and elastic potential in terms of bank accounts where we can save energy, or withdraw it.  Any work done by, or against, external forces are like money being paid into the system from outside (or paid out from the system).  We have no control over this energy, just like we don’t control money that isn’t ours.  The difference between work done against, say, friction, and that done by weight is that we can cash in the energy we’ve ‘lost’ to work done against gravity by moving downwards to release it again.  Just like money isn’t lost when I put it in an ISA in the same way that it is when I spend it in a shop.

As an improvement on SUVAT, I like to use the Felix Baumgartner example again, but this time with energy:

Space Jump Energy

The bungee jump is, of course, a favourite application of elastic potential energy, but we shouldn’t ignore the humble nerf gun (easier to demonstrate in the classroom, too):

Nerf Energy

Once you get to bungee jumping, though, make sure you understand exactly what’s going on at each moment of the jump.  Find a slow-motion bungee jump on youtube, or do what I did this year, and set “watch GoldenEye” as homework the lesson before:

And follow it up with a carefully researched worksheet:

Dam Jumpers

As well as solutions, this worksheet includes descriptions at the end of the different phases of motion: before the rope tightens, while the rope is stretching but before maximum speed, while decelerating to rest, and the bounce back.  It includes values for the energy at each point, as well as details of the forces involved.  For a long time I tried to make a program that would show me exactly what was going on at any given moment during a jump, but – like many energy problems – it becomes a mean differential equation because kinetic is in terms of velocity while gravitational and elastic potential are in terms of displacement.  However, I had a bit of a breakthrough and have made this little GeoGebra beauty in terms of displacement:

Bungee Energy Calculator

I like how it makes that middle section clearer: the brief time while the rope is being stretched but the force isn’t great enough to slow you down yet, only reduce your acceleration.  There’s also an Excel version which will give you more precise figures, but doesn’t illustrate the concept quite so well, I think.

If you enjoyed Top Gear’s Porsche versus Beetle race, you may like to throw this activity in there at the end.  It starts with SUVAT, then improves on the model using energy, assuming a constant air resistance, and then goes further by taking into account variable air resistance (don’t worry – I did the tricky maths already, so there’s a nice-ish formula for students to use).  Although the last bit isn’t necessary, it makes a good extension, as this is what more and more precise mathematical modelling looks like in reality.

Beetle Drop

PS – make sure you watch the race.  You know, to check your solutions:

Circular motion is the next big thing.  As always, start with the definitions and spend enough time messing around with them that the formulae are not only familiar but have practically been derived by students.  I like to start by asking which is going around faster; the Earth around the Sun, or the Moon around the Earth?  Students will, quite correctly, come up with two different answers, depending on their interpretation, and that’s the motivation for a definition of angular speed.

Going Round

Sticking with constant velocity for the moment (but that doesn’t have to be ‘horizontal circular motion’, remember), we can actually calculate the height of a geostationary orbit.  That’s pretty cool, right?

Geostationary Orbit

Don’t be too keen to dive into ‘vertical’ circular motion.  There’s so much fun to be had with the other kind:

Round the BendUp the Wall

(and, to help you out with investigating the Wall of Death further, a useful little Banked Curve Calculator in Excel.)

Finally, we’re at vertical circles.  If your department doesn’t have a loop-the-loop track and a little hot-wheels car, you’ll have to build one.  This is all about getting to grips with the possible scenarios.  A car performing a loop-the-loop might make it all the way round without a hitch, or not quite halfway up, and roll back down, or more than halfway, and fall inwards.  But what happens if the car is fixed to the track, like a roller-coaster?  Or if it’s on the outside of a circular track, like driving over a bridge?  While getting my head around all of this, I made a summary sheet that is as much use to me as it is to the students:

Vertical Circles

Use my GeoGebra illustration to show what happens to the forces as you go around the circle (but bear in mind the details in the previous document, as this demo is limited).

The only other thing to get your head around is that pretty much every single question will involve forces and energy.  Resolve towards the centre, because you know that will be equal to the centripetal force which we have a formula for.  And use conservation of energy, combined with the height, to find the speed.  The link here is the speed.

And today’s homework is to watch Fifth Gear make loop-the-loop work in reality:

And, of course, answer questions on it:

Loop the Loop

The final chapters of M2 take us beyond SUVAT at last.  The progression began back at GCSE (or earlier) with the constant speed equation, was developed further into the constant acceleration equations in M1, and now, in quick succession, using calculus to describe motion when acceleration is given as a function of time, then using differential equations to describe motion when acceleration is a function of displacement.  This opens the doors wide on all sorts of problems, and even though many of them can only be solved using much more advanced techniques, being able to generate the formulae is half the battle in the days of WolframAlpha which will solve your differential equations for you.

The nicest application of these topics, though, brings us right back to circular motion.  It is useful for a teacher to be able to derive the formulae for centripetal acceleration for constant angular velocity.  It’s worth being aware of the derivation for variable angular velocity also, but probably not worth the half-hour it would take to go through it in class!

Deriving Circular Motion

As with M1, I’ve made a Checklist; those key skills which are potential limiting factors for success in M2, but do not, on their own, constitute a full understanding of the module.

The M2 Ideas Test, on the other hand, is often a considerable challenge even for A-grade students, and will really test how comprehensive and thorough a student’s understanding truly is.

And, of course, the M2 Not-Formula Book and the one-page M2 Essentials revision card, both of which summarise key information from the course which will not be provided in the formula booklet, and are highly recommended for independent revision and reinforcement of ideas.

I’d love to hear your thoughts on any of these resources, and of course any others you have found particularly useful.

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FlightPath
A-level Mathematics Resources

“Model the topics as particles connected by light, inextensible strings”: Making sense of A-level Maths: Mechanics 1

All through school I loved algebra for the power it gave me to describe patterns and solve problems.  And I loved physics for the insight it gave me into how and why things work the way they do.

A-level Maths & Mechanics was where they both came together for me.  I can describe archery using algebra, use calculus to explain a car race, even drag trigonometry out of triangles and use it to play conkers.

I’ve been teaching the course for over 5 years now, and the more in-depth my understanding becomes the more I enjoy it.  I love opening the eyes of students to the application of maths they enjoy to real world phenomena that intrigue them.  For a few lessons a week we get that science-y buzz of “now I’ve explained the theory, let me demonstrate it by making this thing spin around / fall off the table / go flying out of the window.”

Below are a few ideas and resources I’ve put together over the years for Mechanics 1.  Watch this space for a follow-up post on Mechanics 2.  This relates directly to our exam board of AQA, but content is broadly similar across the exam boards.

The Language of Maths: The Language of Maths

I like to kick off the year with this little translation activity.  It’s a bit of fun, but it brings out some of the specialist vocabulary used throughout the course, and prompts discussion of assumptions and the mathematical modelling process in general.  Chapter 1 is, on its own, not really examinable, but sets the scene for all of applied mathematics, so in my opinion is well worth a lesson.  I refer back to the mathematical modelling cycle frequently in reference to verifying results, an appropriate level of precision for answers, considering the suitability of assumptions, and so on.

The SUVAT equations (or ‘kinematics equations’, or ‘constant acceleration equaions’) were presented to me as a complete package but I would have loved to see where they came from.  If you’re not quite sure, check out this quick guide – they all come from our basic definitions of speed and acceleration:

Deriving the SUVAT equations

Or use this Equations of Motion investigation with a class to have them generate them for themselves.

Another really handy tool (mostly for teachers, when making up sensible questions, or quickly checking answers) is my SUVAT calculator in Excel.  Comes equipped with unit conversion too, to save you the job:

SUVAT Calculator

On the subject of unit conversions, one of the most useful is the ratio of metres per second to miles per hour.  Why not set this as an exercise for the class?  From 5 miles is roughly 8 km it’s easy to see that 1 m/s is roughly 2 mph.

If students are learning to drive, see if they can derive the stopping distances from the Highway Code.  With a bit of prompting, knowing the distance for one speed should be enough to calculate the rest. A bit of reverse-engineering reveals the standard deceleration rate to be around 6.5 m/s/s.  This worksheet looks at how much more deadly a crash at 80mph would be than a crash at 70mph:

Stopping Distances

The most interesting SUVAT questions involve freefall.  Why did Felix Baumgartner jump from nearly 40,000 metres high?  It wasn’t to give him time to accelerate to the speed of sound (his target); it was to get him far enough beyond our dense atmosphere to reduce air resistance for long enough for him to accelerate freely under the influence of gravity alone.  You can watch his jump here: http://youtu.be/FHtvDA0W34I , but I think I prefer this version:

Which brings us to forces.  When I first taught this I underestimated the confusion that weight and mass can cause, since we use the words interchangeably (often even in the maths classroom) until A-level.  This little fact sheet highlights the differences (and also explains why maths uses g=9.8 while physics still prefer 9.81!):

Weight and Mass

Or for a more in-depth investigation into the force of gravity, see the gravity investigation below, which asks, among other things, why astronauts on the space station, who should experience gravity only fractionally lower than ours, appear to be weightless:

Gravity Investigation

I describe friction as a ‘reactive’ or ‘passive’ force.  Much like the normal (contact) force, it will resist potential motion, or slow it down, but would never, on its own, cause motion.  This idea can be extended in M2 for the contact force when we have circular motion (we must have a centripetal force, and only if there’s no other force providing it, the normal reaction steps up).  I’ve made a simple dynamic illustration of a particle in equilibrium on a slope (and, if you hadn’t spotted it yet, did you know that the minimum coefficient of friction for a particle on an inclined plane is just tan of the angle to the horizontal?)  Friction is the best excuse to tip desks up and watch books slide off.  At 45 degrees?  Coefficient of friction must be 1. It’s worth asking conceptual questions about the friction formula, too.  Why are formula 1 cars designed to have a massive down-draft pushing them into the road?  What forces would change as a result?

Statics is important, and resolving forces for a mechanics student is like taking a blood sample for a medic, or adding VAT for an accountant.  It’s not the hardest thing you’ll ever do, but you’ll need it so often you’d better make sure you not only understand how it works but be able to do it in seconds without breaking your train of thought.   Students will need to get past drawing right-angled triangles for every force so that they can focus not just on a method that works but the best method for this particular question.  Sometimes that’s a vector triangle and cosine rule,  sometimes it’s resolving horizontally and vertically and sometimes it’s picking a better pair of directions to resolve.

Newton’s laws help us to describe how a bicycle can outpace a car – briefly – when the lights turn green, or how a child can tow a 10 tonne barge.  One example I like is dropping something too heavy to hold; instead of just dropping it, let’s say you try to slow its fall.  You can’t completely counteract the force of gravity, but you can reduce the overall downwards force, so that the object accelerates downwards less quickly and you may avoid a broken washing machine.

Hold On

Projectile motion is a great application of 2-D SUVAT equations, but I often like to solve the problems in two separate sections – horizontal motion on the left, with its nice constant speed equation, and vertical motion on the right, using SUVAT under gravity.  The only connection between horizontal and vertical motion is the time t.  This trebuchet worksheet encourages students to deal with more than just known speeds and angles.  And you’ll know you’ve properly grasped projectiles when you can explain why doubling the speed quadruples the range.  This next worksheet brings in a non-zero initial height (inspiration drawn from my time on board ship, where heaving lines are thrown from the deck to the quayside):

Projectiles Worksheet

For an easier, concept-based introduction to parabolic motion (perhaps as a homework prior to learning the maths behind it), try FlightPath:

FlightPath actifity

Finally, I’m going to recommend my M1 Checklist (a list of all the basic skills that, if lacking, will cause problems), the M1 Ideas Test (great for ironing out misconceptions and going beyond ‘can solve problems’ to ‘understands the concepts’), and my M1 Not-Formula Book and the one-page M1 Essentials revision card, both of which summarise key information from the course which will not be provided in the formula booklet, and are highly recommended for independent revision and reinforcement of ideas.

More Not Formula Books and Essentials Cards are available from my website:

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