brightsky's Maths Thread

[lzxnl]

2. As a space shuttle moves through space, its mass decreases due to fuel consumption. At time t, the mass of the shuttle is m kg (a scalar), its velocity is v m/s (a vector), its acceleration is a m/s^2 (a vector), and the net force acting on the shuttle is F N (a vector).

Well I'll give you a heads up on how rocket equations work.
For starters, we need a rate of fuel consumption. It's convenient that the rate of mass consumption = dm/dt

Assume that at t = 0, the rocket and unburnt fuel has a certain mass m + dm and velocity v, where dm is the mass the rocket is about to lose.
Then in time dt, as the rocket ejects some gas with speed ve, its momentum drops by dm*ve.
So dp for the rocket = -dm*ve as the momentum of the rocket must increase (think about it...conservation of momentum, you're pushing something back), and dm is negative.
Assuming that the mass is ejected with the same speed
Change in momentum for the rocket = m dv/dt = -dm/dt * ve
So m*a = -rate of mass ejected * velocity mass is ejected
Or m dv = -ve dm
Separable equation, you can solve for v in terms of m.

Unfortunately that doesn't really answer your question. But you see the sort of info that you need to be able to answer the question.

[brightsky]

Quick question:

A particle is moving so that its position at time t seconds is given by r(t)=2cos(pi/10*t) i + 3t*j. Find the Cartesian equation of the path of the particle. State the domain and the range of the path.

So...is the range [0, infinity] or [0, 30]? And how would you express the Cartesian equation of the path? If you write cos^-1(...), technically you're already artificially restricting the domain. Should I write something like Cos^-1(...)?

[lzxnl]

So y=3t
x=2cos(pi t/10)
=2cos(pi*3t/30)
=2cos(pi*y/30)

That's a Cartesian equation of the path.

[nerdgasm]

Hmm, I think that the range would be [0, infinity) (making sure to use the round bracket when using -infinity or infinity in your domain ).

Why? The way I like to look at it, is to analyse the motion in the x-axis (controlled by the i component) and the motion in the y-axis (controlled by the j component) separately.

For your motion in the x-axis, your particle basically oscillates from 2 to -2 and back again, much like an ideal spring would. So it'll just go back and forth repeatedly, achieving its maximum speed at 0, and its minimum speed at 2 and -2, due to the effect of the 'cos' in the expression of i. For your motion in the y-axis, your particle just travels upwards at a rate of 3 units per unit of time. But there is no reason at all why the particle has to stop after going from 2, to -2 and back to 2 again. So there isn't really a restriction on how 'positive' the range can be.

I like nliu1995's approach to this question; finding x in terms of y eliminates a lot of hassle. With regard to your query though, MathsQuest says cos^-1(x) and Cos^-1(x) are the same thing and Wikipedia says they are merely separate conventions. I honestly can't remember what is required for VCAA purposes, (which is probably what is most important at this point in time), but it seems they say cos^-1(x) already has the restricted domain in their formula sheet.

[brightsky]

So y=3t
x=2cos(pi t/10)
=2cos(pi*3t/30)
=2cos(pi*y/30)

That's a Cartesian equation of the path.

Oh yes...I guess that's a better way of expressing it, although I get the feeling that VCAA prefers that we express y explicitly. The problem with saying that the range is [0,30] is that you're artificially imposing that restriction yourself. Nowhere in the vector equation does it tell us that y must be restricted in this way.

[lzxnl]

This is why I saw screw VCAA.

[brightsky]

So:

A one-dimensional space in R^n is called a line in R^n.
A two-dimensional space in R^n is called a plane in R^n.
A (n-1)-dimensional space in R^n is called a hyperplane in R^n.

But what is the technical name for a three-dimensional space in R^5?

Thanks!

[Alwin]

So:

A one-dimensional space in R^n is called a line in R^n.
A two-dimensional space in R^n is called a plane in R^n.
A (n-1)-dimensional space in R^n is called a hyperplane in R^n.

But what is the technical name for a three-dimensional space in R^5?

Thanks!

Hmm, I'm not entirely sure either.
In MUEP, (when talking about injective & subjective matrices) the lecturer called it a mapping of R^3 into R^5, suggesting R^3 is a subspace of R^5.
In UMEP, (when talking about vector spaces) my teacher calls it a vector space in R^5.

So, if I had to call it something, I'd say a three-dimensional space in R^5 is just a subspace (or vector space) depending on the q.
Not 100% either though . . .

[brightsky]

1. Let A be a m*n matrix. Is it true that nullity(A) = nullity(A^T)?
2. Let A and B be n*n matrices. S is the set of all n*n matrices M that satisfy the equation AM + M^TB = 0. Is S a subspace?

EDIT: Got the second question; 'twas easier than I thought. Still stuck on the first though...

[Lasercookie]

1. Let A be a m*n matrix. Is it true that nullity(A) = nullity(A^T)?

Not in general. A simple way I can think of doing this is to show (if you haven't already done this one before, think about the relationship between the dim col(A) and dim row(A)) and then just use the Rank-Nullity theorem.

Nullity(A) = n - Rank(A) = n - Rank(A^T)
Nullity(A^T) = m - Rank(A^T) = m - Rank(A)

You can see that Nullity(A) = Nullity(A^T) for a square matrix (since m = n), but not for a non-square matrix.

[kamil9876]

Most trivial example would be A=[0 0]. Nullity of A is 2 while it is 1 for A^T.

[brightsky]

Ahh...of course! Thanks so much laseredd and kamil!

[brightsky]

An elastic string has modulus of elasticity 2g N. If a 500 g mass hangs vertically from the string, the string extends 20 cm from its natural length. The natural length of the string in cm is?

Can someone give me a rundown on simple harmonic motion? Thanks!

[lzxnl]

An elastic string has modulus of elasticity 2g N. If a 500 g mass hangs vertically from the string, the string extends 20 cm from its natural length. The natural length of the string in cm is?

Can someone give me a rundown on simple harmonic motion? Thanks!

I'm fairly sure your units for the modulus of elasticity are incorrect. Isn't the elastic modulus stress/strain, where strain is dimensionless and stress is units of force?

[brightsky]

1. Calculate the dimensions of the image and kernel of the linear transformation T:R^3 -> R^3 by reflection in the plane x+y+z=1. Is T even a linear transformation? If we reflect (0,0,0) in the plane x+y+z=1, it becomes something else...
2. T(x,y) = (2x+y,x+y,x-y,x-2y). Explain why the linear transformation T^2 isn't defined. I just wrote: matrix multiplication doesn't work, but I don't feel as though that is adequate.
3. What's the transformation matrix for a reflection in the line which makes an angle theta with the positive x-axis? I was thinking [cos2x, sin2x; sin2x, cos2x] but I don't think that's right...

Thanks!