Inspired by Aarsh Verdhan

Classical Mechanics Level 5

A $1 kg$ particle exists in a 1-dimensional coordinate system. At time $t = 0$ , the particle is at rest at $x = 0 \, m$ . It experiences a force (in Newtons) described by the following equation:

$\large{F(x,t) = 3xt + 1}$

At what time (in seconds) does the particle reach $x = 4 \, m$ ? Give your answer to 2 decimal places.

The answer is 1.85.

1 solution

Steven Chase
Apr 1, 2017

This problem seems to be either Level 6 or Level 2, depending on how you do it. It seems to be described as an Inhomogeneous Airy Equation , the formalities of which I am not familiar with. However, the numerical solution is quite trivial, as shown below. I was curious to see if anybody would come up with an analytical solution.

m = 1.0

x = 0.0
v = 0.0

t = 0.0

F = 3.0*x*t + 1.0
a = F / m

dt = 10.0**(-6.0)


while x <= 4.0:

    x = x + v * dt
    v = v + a * dt

    F = 3.0*x*t + 1.0
    a = F / m


    t = t + dt

print t

The "Level 6" approach involves finding two distinct power series solutions for the homogeneous equation, then calculating the Wronskian and applying variation of parameters to find the particular solution. I gave up after the first step, since calculating the Wronskian would have been an unholy mess. Numerical methods are definitely the way to go here.

Brian Charlesworth - 4 years, 2 months ago

Yeah, that's what I figured. But you didn't choose to solve?

Steven Chase - 4 years, 2 months ago

No, I had spent quite some time trying to solve it analytically, and when I saw that a solution had been posted, my curiosity to quickly see what method you, (or assumed to be you), had come up with outweighed my desire to get an actual solution value.

Brian Charlesworth - 4 years, 2 months ago

@Brian Charlesworth – Ok, cool. Hopefully it wasn't too much of a waste of time. There seems to be a general dichotomy in terms of how to solve problems.

Method 1: Sophisticated analysis with very little computation
Method 2: Very simple analysis with lots of computation to take up the slack

I find this dichotomy to be fascinating. I have a suspicion that the actual machinery of the universe is very simple (akin to Method 2), with a lot of "computation" backing it up. Our fancy mathematical abstractions may just be ingenious human tricks which (up until the modern age) were essential in order to compensate for our lack of calculating ability. But then again, the mathematical constructs are often of such beauty that it is hard to avoid thinking that they might have some deeper meaning.

Steven Chase - 4 years, 2 months ago

@Steven Chase – Not a waste at all. The DE looked mostly harmless at first glance, but after dealing with the homogeneous equation I took a dive into an old text to figure out what to do for the particular solution and then realized how difficult it was going to be. I have a stubborn tendency towards Method 1, but I hit a wall on this one. :)

Rather than a dichotomy, I see more of a continuum between the two paradigms. Doing as much analysis as one can before the returns on your efforts diminish, (which depends on individual tolerance to be sure), and only then resort to computational methods, seems like the ideal balance. But that's just about process; as for the "actual machinery" of the Universe.... Call me Platonic, but I do put great stock in the beauty of mathematics, and the more elegant it is the more closely it mirrors the Universe. This belief isn't absolute though; as Feynman said: "We are not to tell nature what she's gotta be .... she always has a better imagination than we have". Also, specific problems can have so many variables in play that the only reasonable way to deal with them is through technological means.

Brian Charlesworth - 4 years, 2 months ago

Inspired by Aarsh Verdhan

The answer is 1.85.

1 solution

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