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Grizabella Offline
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Matrix/linear algebra + physics - September 24th 2009, 10:24 PM

We've got this assignment in my matrix algebra class, and it's been going pretty well, but I'm having trouble with the last question, since I think it's related to physics, which I haven't been taking. I've managed to do the questions on applications of matrices to economics, chemistry and urban planning, but the physics one, I have no idea where to start. Could anyone give me some tips on how I should go about solving this question? I don't really know how to put in vectors on here, so I'll do my best to describe them.

The momentum P (with vector sign on it) of a system of n particles in space with masses m1, m2,...,mn and velocities v1, v2,...,vn (the velocities are vectors) is defined as

P = m1v1 + m2v2 + ... + mnvn

Now consider two elementary particles with velocities
v1 = (a 3x1 matrix, with numbers 1, 1, 1) and v2 = (a 3x1 matrix, with numbers from top to bottom 4, 7, 10)

The particles collide. After the collision, their respective velocities are observed to be

w1 = (3x1 matrix, 4, 7, 4) and w2 = (3x1 matrix, 2, 3, 8)

Assume that the momentum of the system is conserved throughout the collision. What does this experiment tell you about the masses of the two particles?

The only thing I can think is to solve m1v1 + m2v2 = m1w1 + m2w2, but I'm really not sure that's what it wants.


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Re: Matrix/linear algebra + physics - September 25th 2009, 01:02 AM

Yes, you are on the right track. Solve for m1 and m2 by moving stuff around in that equation. You will be able to factor the masses out and get an expression of the form:

m1*(some difference in velocities) = m2* (some other difference in velocities)

Since the velocity vectors are given, you can easily use matrix subtraction to calculate the difference. So then you will have something like:

m1*(a new velocity) = m2* (another new velocity).

Now since mass is a scalar quantity and not a vector, you can simplify things by taking the magnitude of each of those new velocity vectors you obtained. Then you will have:

m1*(some number) = m2*(some other number)

Now you can solve for a ratio of masses (either m2/m1 or m1/m2, doesn't really matter). The value of that ratio will tell you which mass is larger. For example, if m2/m1 > 1, then m2 > m1.

I hope this helps.
   
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