# Rigorous Dimensional Analysis at the DfE

OK, I’ll warn you now. Some of you may think what follows is pedantry, pure and simple. For that I make no apology (but I do make an apology for spelling Rigorous wrong in the original title of this post – the irony is not lost on me).

Yesterday, the GCSE subject content for Combined Science was published and (as brought to my attention on Twitter by @Alby) on page 35  were the set of physics equations that students were expected to learn off by heart – they included one that I had never seen before:

kinetic energy = 0.5 × mass × (acceleration)2

Now being a trusting chap I thought they must know something I don’t so I thought I’d bring some dimensional analysis to bear on this equation. This is a technique you can use to check that the units (the dimensions) of an equation match on both sides; if they don’t then Huston we have a problem.

So kinetic energy is equivalent to work done  which (as mentioned on the same page) is equal to  force × distance.  Time to bring in the units: newtons for force and metres for distance.

Force as we all know (ahem – see the ResearchEd video coming soon for my own quickly corrected equation boob) equals mass × acceleration. Acceleration has the units metres / second / second or metres / second2   so the units for force can be expressed as kilograms × metres / second2 .

Therefore the kinetic energy which  is usually expressed with the units joules can also be expressed in terms of kilograms × metres / second2 × metres or kilograms × metres2 / second2

So for the equation quoted from the DfE list above is to be a valid equation the units (or dimensions) on the right hand side of the equation have to work out the same; lets see:

0.5 is dimensionless number, so doesn’t contribute any units. Mass gives us kilograms, and accelerationhas the units ( metres / second)2. Combining these gives dimensions of kilograms × metres2 / second4 which is not what we had on the left hand side.

So this equation is clearly not going to pass mustard (yes I know) with my Y10s when I’m teaching the new GCSE curriculum them in 2 years time.

OK so this is an easy mistake to make (well not really – how anyone with any acquaintance with physics typed that equation out is beyond me) but then they did it again.

I’ll not bore you with the maths again, but at the bottom of the page was

(final velocity)2– (initial velocity)2= 2 × acceleration × time

This equation is supposed to be one of the equations of motion that are used to calculate changes in motion during uniform acceleration. However, it is also wrong. It should be:

(final velocity)2– (initial velocity)2= 2 × acceleration × displacement

with both sides having the dimensions metres2 / second2

The DfE have since changed the document (but not before Richard Adams posted this comment in the Guardian ) and republished it in Word format – presumably so they can quickly fix any other mistakes.

They’ve corrected it right, so what’s the problem? They made a couple of mistake in some rather fundamental physics equations and sent them out as guidance. They would have been picked up, undoubtedly, by the exam boards making the changes to specifications if not before.

But their amendments still aren’t correct; they refer to distance in that second equation when it should be displacement, a vector. They refer to speed in the KE equation when really it’s the dot product of velocity which is a vector (the students are supposed to know about the difference between vectors and scalars so why doesn’t their equation list reflect this).

I’ve not even mentioned (though I will now) the other bloopers on that page  such as referring to as the gravity constant when it is the gravitational field strength (nominally 9.81 m/s2 or N/kg) and it is anything but constant (the gravitational constant G is a completely different number 6.67 × 10-11m3 kg-1 s-2).

Using the term charge flow in an equation alongside current is nothing short of confusing but I’m perhaps being a little over zealous here.

And efficiency = output energy transfer / input energy transfer will only ever result in an efficiency of  100% as the energy is conserved – they mean to say the useful output energy transfered

And  it’s not the first time; last year they mistakenly quoted a definition of Newton’s Second Law as his Third in the KS4 Programmes of Study draft.

And this is just the Physics.

I completely understand that people make mistakes – I make plenty, but these are so obvious that they should have found by proof readers in the DfE or their consultants not by physics teachers on Twitter during their Easter break (almost).

If I was feeling snide I’d mention something about the rigour of the new GCSEs, but I’m not, so I won’t.

I will however repeat my lack of apology for pedantry and I offer my proof reading services to the DfE for when they next release a set of equations.

Here’s the updated document

https://www.gov.uk/government/publications/gcse-combined-science

and Alby’s original tweet:

Edit: Other people who spotted and pointed out errors in this list include @DrDav, @HRogerson and @miss_m_w
Alby has been quoted in this story by Channel 4 News