Our new WitCH, below, comes courtesy of Charlie the Enforcer. Once again, this WitCH is from the 2018 SCSA Mathematical Methods Exam (here and here): it’s the gift that keeps on giving. (And a reminder, WitCH 2 and WitCH 3 still require attention are still unresolved.)
Question 11 and the solution in SCSA’s marking key are below. Happy hunting.
Update
John has pretty much caught it all. The killer issue is the use of the term “deceleration” in part (c) which, the solution implies, refers to the drone speeding up in the southerly direction. This is arguably permissible, since deceleration can be (though is far from universally) defined as a negative acceleration, and since way back in part (a) it was implied that North coincides with the positive x direction.
Permissible acts, however, can nonetheless be idiotic: voting Liberal or Republican, for example. And, to use “deceleration” on a high stakes exam to refer implicitly to increasing speed is idiotic. Moreover, to use “deceleration” in this manner immediately after explicitly indicating the “due south” direction of motion is truly ruly idiotic. Still not as idiotic as voting Liberal or Republican, but genuinely special-effort idiotic.
That’s enough to condemn the question, even by SCSA standards. But, the question is also awful in many other ways:
- The question is boring and butt ugly.
- No indication is given whether exact or numerical solutions are permitted or required.
- Having a drone an arbitrary 5m up in the sky for a 1D problem is asking for trouble. For example:
- The “displacement” of x(0) = 0 for a drone 5m up is pretty stupid.
- “Where is the drone in relation to the [mysterious] pilot?” Um, kind of uppish?
- “How far has the drone travelled …” is needlessly wordy and ambiguous. If you want a distance, for God’s sake say “distance”.
- Given the position function x(t) is at hand, part (c) can easily and naturally be solved by hand. But of course why think about things when you can do mindless calculator crap?
I’ll make a couple of quick observations (while no doubt missing the main point) since no-one else has jumped in yet:
1. The displacement x is not defined properly. You can fly due North or due South but still be moving up and down while doing so …. There is an assumption that due North means in a straight line. One could argue that the expression given for v implies a straight line. Nevertheless ….
2. “Ava is flying [the] drone …” is stated at the start of the question, part (b) refers to the “pilot” and part (c) refers to Ava again.
Switching from “Ava” to “pilot” to Ava is poor wording and could easily create confusion (particularly among ESL students).
3. The drone passes “… directly over [Ava’s] head …” and so directly above Ava’s head has implicitly been defined as the origin. So “Where is the drone in relation to [Ava] …” doesn’t really make sense to me. The answer “… (27 cm) due South of [Ava].” makes no sense to me because the origin is directly *above* Ava (somewhere) …..
4. In part (c) I would have preferred ‘Find the distance travelled from t = 0 ….’ rather than the wording used.
All in all, I found the question clumsy and confusing. Lucky I wasn’t sitting the exam!
Thanks, John. Not the killer issue that Charlie the Enforcer flagged, but all valid points. Definitely the wording is poor, particularly the mission of “distance” in part (c). And yes, the entire 1-D framing for a drone 5m in the air is appalling. But, there’s worse there.
I find the use of modulus signs inside the integral in the solution a bit strange, but haven’t yet worked through the problem to see if this causes any issues. The implicit definition of the origin is pure crap though.
Getting distance from v = v(t) by integrating |v| (wrt t) is OK.
From a v-t graph:
The signed area (integral of v wrt t) gives displacement.
But he magnitude of the area gives distance. So if you take a |v|-t graph, all the negatively signed area becomes positive, so integral of |v| (wrt t) gives distance.
Yes, the calculation of the distance traveled to t = 10-ish is valid. Is it smart?
OK, using the word deceleration really, really bugs me. I’ve never liked the word. At t = 10-ish, the drone is *accelerating* towards Ava. It’s *speed* is increasing. It’s only decelerating in the sense that its acceleration is negative because the Northly direction is positive ….
The question should have asked for distance travelled by the drone when its acceleration was -0.5 m/s^2.
Anyway, enough with my personal gripes. With regards to whether the given solution is smart. There’s a much better calculation that gives the exact answer:
You have
x = -6Cos[t/3 + pi/6] + 3Sqrt[3] …. (1)
from part (a). From the given acceleration, you have -0.5 = 2/3 Cos[t/3 + pi/6] therefore
Cos[t/3 + pi/6] = -3/4 …. (2)
Sub (2) into (1): x = 9/2 + 3Sqrt[3]. This is when the drone is slowing down and travelling North.
From (1), the drone momentarily stops at x = 6 + 3Sqrt[3], which is a distance of 3/2 from 9/2 + 3Sqrt[3].
So the drone will have an acceleration of -0.5 again (but speeding up) after travelling an exact distance of 9/2 + 3Sqrt[3] + 3 = 15/2 + 3Sqrt[3] metres.
The question does not specify an accuracy for the answer. Any reasonable person would assume that means an exact answer is required. But the answer key gives a method that can only ever give an approximate answer – and in this case three decimal places is randomly chosen …. Will they accept the less accurate 12.7 metres? 13 metres? Where do you draw the line …?
I give the marking key for part(c) *zero* marks because:
(1) It uses a method that will only ever give an approximate answer (so no method marks)
(2) It does not give the correct final answer, namely the exact answer of 15/2 + 3Sqrt[3] metres.
Although the question is lame and I have gripes with the wording, the marking key is the real crap here. And it stinks.
Bingo. The question is foolish and foolishly worded from start to end, but the use of the word “deceleration” is fatally ambiguous. It is impossible to determine the intended meaning of the word.