Problems 13/03/13

There are way too many problems for us to be able to go through during the session. Therefore, please try them at home first!

1. Wedge (source: maths classes)

There is a marble of mass m lying on a wedge of mass M. The marble slides down the wedge

frictionlessly and therefore causes the wedge to move (frictionlessly). Find acceleration of the

wedge in terms of m, M, g and α

2. Tsiolkovsky's rocket equation (source: practice interview)

Imagine a rocket flying horizontally. It burns its fuel at a constant rate to gain velocity. Using

momentum equation try to prove that the change of velocity is ΔV =ve ln(m0

m) , where ve is

the exhaust velocity. Hint: you must use mathematical modeling with calculus!

3. Classic: sketching! (source: interview at Cambridge and (c) from my practice interview)

(a) x2= k 2

1−e− y

(b) y= x3

1+x2

(c) draw graphs of acceleration, velocity and displacement against time of a water bottle

4. Find all the points such that: (source: interview at Cambridge)

−43π

x+2=cos (x )

5. Tricky maths questions (source: interview at Imperial)

(a**) Is ∑r=1

∞ 1r 2

divergent or convergent? Prove your answer.

(b *) What about ∑r=1

∞ 1r

?

(c) After solving that, try to prove that ∑r=1

1000 1r<10 without a calculator.

(d) Investigate ∫0

3 1(1−x)2 dx

(e) Prove that n3−n is divisible by 6 for all n∈Z

6. Some difficult mechanics problem (source: practice questions from Trinity College)

One end of a rod of uniform density is attached to the ceiling in such a way that the rod can swing

about freely with no resistance. The other end of the rod is held still so that it touches the ceiling as

well. Then the second end is released. If the length of the rod is l metres and gravitational

acceleration is g ms−2, how fast is the unattached end of the rod moving when the rod is first

vertical?

7. A qualitative modeling question:

Why does a sausage always break on heating vertically rather than along the circumference?