# Projectile motion with resistance proportional to the square of the velocity (1 Viewer)

#### tywebb

##### dangerman
The syllabus is unclear as to whether this is required.

Even some textbook authors have contacted NESA to gain clarity on this issue to no avail.

Some texts include it. Some don't.

In particular, Cambridge and Terry Lee don't.

New Senior Maths and Maths in Focus do.

You may be wasting your time doing it if it is never going to be examined.

A former HSC chief examiner once commented at an HSC feedback day that throughout the life of a syllabus, every aspect of the syllabus should be examined at least once.

So unless we get more clarity from NESA as to the expectations in regards to this, then we will potentially only really know when the last HSC exam is sat for the NEW syllabus!

Of course we may know before that if there is a question on it before then.

But I think that this way of letting people know what is required is totally unacceptable.

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#### Trebla

Why does it matter though? Does it require the learning of new concepts or just applying existing concepts already mentioned in the syllabus (i.e. deriving the motion equations)?

My interpretation of the new syllabus dotpoint is that students should be able to handle any resisted projectile motion problem, so the type of the resistance force can be anything - as long as students apply the force concepts needed to derive the required equations.

If textbooks want to cover specific types of resistance forces on a projectile in a chapter then that's really up to them. It's no different to how some textbooks dedicated a chapter to properties of definite integrals even though it's not explicitly called out in the syllabus (as it doesn't really use any new concepts - just applies existing ones).

#### tywebb

##### dangerman
Why does it matter though? Does it require the learning of new concepts or just applying existing ones (i.e. deriving the motion equations)?

My interpretation of the new syllabus dotpoint is that students should be able to handle any resisted projectile motion problem, so the type of the resistance force can be anything - as long as students apply the force concepts needed to derive the required equations. If textbooks want to cover specific types of resistance force in a chapter then that's really up to them. It's no different to how the old syllabus allows coverage for anything under the Harder 3u topic, yet some textbooks decided to dedicate a chapter to harder probability (even though it's not explicitly called out in the syllabus).
Well Cambridge don't just not do it. They also say why they don't do it.

Here is what they say about it (on page 257):

"There are also some questions involving projectile motion with air resistance, however, these are very limited as solutions can only be found in one case, when the resistance is proportional to the velocity."

Going beyond this is of course possible, but you then very quickly end up in a situation which is WAY beyond the syllabus - even for Extension 2.

Any pretence that it is OK to do it within the syllabus will inevitably lead to unacceptable compromises.

This is in fact what has appeared in New Senior Maths and Maths in Focus.

#### Trebla

Well Cambridge don't just not do it. They also say why they don't do it.

Here is what they say about it (on page 257):

"There are also some questions involving projectile motion with air resistance, however, these are very limited as solutions can only be found in one case, when the resistance is proportional to the velocity."

Going beyond this is of course possible, but you then very quickly end up in a situation which is WAY beyond the syllabus - even for Extension 2.

Any pretence that it is OK to do it within the syllabus will inevitably lead to unacceptable compromises.

This is in fact what has appeared in New Senior Maths and Maths in Focus.
How do New Senior Maths and Maths in Focus approach it when they mention it?

It is possible that the questions could be limited to say finding the acceleration equations, not solving all the way to the displacement equations. It's like the integral of $\bg_white e^{-x^2}$. The syllabus doesn't explicitly say to exclude integrating $\bg_white e^{-x^2}$ but you can certainly ask questions relating to the integral $\bg_white e^{-x^2}$ without directly integrating it (e.g. approximations by Simpson's rule or using properties of even functions).

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#### Carrotsticks

##### Retired
I wouldn't make it part of my standard teaching lessons. More like an add-on for the times when you get those few students who just find the linear case too easy and you want to keep them busy otherwise they'll find cheeky ways of spending their energy like distracting others.

Integrating this into the school lesson will waste valuable time that can be better spent elsewhere. By addressing the square case, the focus is drawn away from the actual concept (resisted projectile motion) and more towards the complicated integration and algebraic techniques involved.

So would I ever cover it? Absolutely yes, if the cohort can handle it. Would I force it on all of my students as a 'standard requirement'? Absolutely not.

#### blyatman

##### Well-Known Member
Yeh I'm with Trebla on this one. I don't see why the resistance being proportional to the square of the velocity is relevant. They should be able to solve any (reasonable) resisted motion problem, whether its directly proportional to the velocity, proportional to the square of the velocity, or whatever else. The concept is still the same, the only difference is the algebra. Didn't the old MX2 syllabus have resistance proportional to the square of the velocity, where you just had to solve it using partial fractions?

#### tywebb

##### dangerman
Yeh I'm with Trebla on this one. I don't see why the resistance being proportional to the square of the velocity is relevant. They should be able to solve any (reasonable) resisted motion problem, whether its directly proportional to the velocity, proportional to the square of the velocity, or whatever else. The concept is still the same, the only difference is the algebra. Didn't the old MX2 syllabus have resistance proportional to the square of the velocity, where you just had to solve it using partial fractions?
That was motion in a line. Much easier to do with resistance proportional to square of velocity.

What I am talking about is projectile motion with resistance proportional to the square of the velocity. That's totally different and much harder to do - even when compromises are made such as the ones in some textbooks. If you don't make those compromises then the solution is beyond the scope of the syllabus.

#### blyatman

##### Well-Known Member
That was motion in a line. Much easier to do with resistance proportional to square of velocity.

What I am talking about is projectile motion with resistance proportional to the square of the velocity. That's totally different and much harder to do - even when compromises are made such as the ones in some textbooks. If you don't make those compromises then the solution is beyond the scope of the syllabus.
Ah right, yeh fair enough. However, you can solve the individual x(t) and y(t) components in a relatively straightforward manner. The difficult part is getting the Cartesian form y(x). But yeh realistically speaking, it's probably safe to assume it won't be assessed.

#### tywebb

##### dangerman
Here are more details from new senior maths:

#### Trebla

My understanding is that the direct analytical solution is actually impossible to derive, so obviously they won't ask for it in the exam.

From my reading of this, the only part that is 'outside' the syllabus is the "let's make some simplifying assumptions" part. The rest is just algebra bashing all within scope of the syllabus. I wouldn't expect students to be able to call upon these assumptions, but rather it be given to them in the question.

In my opinion, the textbook is addressing this case purely at their own discretion. It is not a necessity to "memorise" for the syllabus but that doesn't rule it out from being asked in the exam (with some guidance).

#### tywebb

##### dangerman
The problems in MIF and NSM textbooks stem from the incorrect assumption that -kv2 can be split into horizontal and vertical components -k(vx)2 and -k(vy)2.

Steve Howard has provided the following explanation as to why this cannot be done.