Natural Sciences Interview #2

Part 1 is here.

Interview 2 – physics and maths

  1. Have you heard of Felix Baumgartner?

Yes, he jumped off some balloon.  I was informed that he had jumped from 40,000m.

  1. Tell me about the forces involved in his jump. 

Obviously, the force of gravity, and the force of air resistance. I said that I couldn’t find his equation of motion, because I didn’t know about the air resistance. We came back to this shortly.

  1. What value of g are you using?

I immediately perceived that the question was about the variation of g. I wrote down g=GM/R2. I knew that the radius of the earth was about 6400km, so I did a quick mental calculation to note the difference in order of magnitude between 40,000m and 6400km. Therefore, I concluded that the variation of g was not relevant. He agreed, but asked me to quantify it.

I replaced R with R+h, (R is the Earth’s radius and h is the height above the surface) and expanded, but didn’t know how to proceed.

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I think the key point was that the h2 term vanishes in comparison to the R2 and the 2Rh.

  1. How does air pressure vary as a function of height?

This was quite frustrating. Everyone knows that there is an inverse relationship, but I had never stopped to consider the exact function.

I said that I could work it out if I knew the distribution for pressure or density.

  1. Have you heard of the Boltzmann distribution?

Yes, it is a distribution for the velocities of particles in an ideal gas. I don’t know the precise form of the equation, except that it has an exp(1/T) in it.

This was a weird question, and seemed a bit out of place. But it related to question 4. Since I didn’t know the form of the Boltzmann distribution, the guy just said “assume that the pressure halves every 5000m [the number is irrelevant I think]”.

I knew exactly where he was going, so intuitively wrote  \rho = \rho_0 (\frac{1}{2})^{h/5000}. He seemed happy with this, so we left it there.

  1. With the above in mind, could Felix have broken the sound barrier?

 I thought that he wanted me to use the pressure expression to do a differential equation (I was all pumped and ready for this). But when I started, the interviewer got a bit impatient and told me that the pressure thing was a separate part of the question. I think that was a bit unfair, but never mind.

What they wanted was much more simple. I had already shown that changes in g were pretty much irrelevant. Likewise, because of the exponential relationship of air pressure, air resistance was also negligible.

So it was a simple freefall suvat equation. I did the suvat in my head, and the guy was surprised I got the right answer.

 [second interviewer takes over]

  1. f(x) = \arcsin (\cos x) . Differentiate this function

 I started off by saying that I would simplify the function. This is because I got f(x) confused with something similar, cos(arcsinx) [which I know how to simplify]. So I didn’t know how to simplify it, and had to backtrack and say I would just differentiate it normally.

Easy chain rule, and as usual, I skipped  few steps (which I later regretted).

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  1. Find the definite integral between 0 and 1 of f(x)

I had absolutely no clue how to integrate it (at first). I started off by suggesting substitutions, but knew that these wouldn’t work.

Luckily I had a mini ‘eureka’ moment. Look at the derivative, it’s a negative constant. This means that the f(x) ALWAYS has the same negative gradient – it’s just a straight line.

So I knew that f(x) could be rewritten as –x + c. I didn’t know how to proceed at first, but the interviewer suggested “how might you find c…” and I worked it out. c = f(0) = \frac{\pi}{2}.

The integral is just the area of the triangle. The interviewer seemed extremely happy that I had managed to spot this method.

  1. You missed out a step, what did you miss out?

As soon as he said I missed a step, I knew where my error was, but it took me a while to figure out exactly what it was. I knew that I had messed up the square root of (1 – cos2x) in the derivative. I suggested that I had forgot a +/-, but it wasn’t that. We then had a really sad discussion (sad and embarrassing for me) about square roots. 

  • What is the square root of 16?

+/- 4, but the sqrt sign technically means only the positive root, so 4.

  • What is the square root of 4?

2.

  • What is the square root of ( -2)2?

I got really flustered, said -2, then quickly corrected to 2.

  • What is the square root of x2?

Thank goodness I realised what he was getting to, the answer is clearly |x|. I’m ashamed it took me so long.

  1. What is the factorial of a half? 

I smiled and said sqrt(pi)/2. (I mentioned the Gamma function in my personal statement).

  1. In your maths exploration, how did you show that the gamma function worked as a factorial?

Repeated integration by parts. I asked if he wanted me to do it, he said no and seemed to be content with this answer.

  1. Is the Gamma function the only function that can model a factorial?

Obviously not. Given any (finite) set of coordinates, there are an infinite number of functions that can model it.

However, I said that I had proved that the gamma function satisfies the recursion principle, and that gamma(1)=0. This means that it is at worst, a superset of the factorial. I also said that I was aware that the Gamma function had an extra property which meant that it was the only function that satisfactorily extended the factorial to non-integers, but I was honest and said I didn’t know exactly what it was.

He told me that the name of the property was analytic continuation. Thanks mate.

  1. Why do you want to study natural sciences?

 In my personal statement, I had already talked about the philosophical aspects – I love to find simple explanations, reductionism etc. I hinted at that in this answer, but ended up saying “these are where my strengths lie”. He said “so you’re basically saying that you’re good at this subject”. I gave a cautious “yeah”

14. How are you going to keep your academics up to scratch prior to entering?

 I said that I had already planned which books I would study, mentioning the Feynman lectures. They were happy, and said that it was a good book.

  1. Any questions? You don’t have to try to impress us with good questions, we’re genuinely asking if you have any questions.

I asked something administrative about UCAS.

THE END

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