Dynamics Question

[epiphany]

My hw problems looks like this:

The acceleration of a particle is directly proportional to the square of the time t. When t=0, the particle is at x=36 ft. Knowing that at t=9 s, x=144 ft. and v=27 ft/s, express x and v in terms of t.

I'm not exactly sure what directly proportional means. Can someone help me out?

Also I have another question I'm stuck on:

The acceleration of a particle is defined by the relation a= 12x-28, where a and x are expressed in m/s2 and meters, respectively. Knowing that v=8 m/s when x=0, determine (a) the maximum value of x, (b) the velocity when the particle has traveled a total distance of 3m.

[Tom Mattson]

I'm not exactly sure what directly proportional means. Can someone help me out?


MathWorld is your friend.

http://mathworld.wolfram.com/DirectlyProportional.html


Also I have another question I'm stuck on:
The acceleration of a particle is defined by the relation a= 12x-28, where a and x are expressed in m/s2 and meters, respectively. Knowing that v=8 m/s when x=0, determine (a) the maximum value of x, (b) the velocity when the particle has traveled a total distance of 3m.

Note that a=\frac{d^2x}{dt^2} and solve the resulting differential equation for x. You should be able to answer the questions using that solution.

[epiphany]

Sorry, I'm kinda slow. I'm still not sure what they mean in the question with directly proportional. Does it mean that a = t squared?

[Benny]

a = 12x - 28 \Rightarrow \frac{{d^2 x}}{{dt^2 }} - 12x = - 28


You have a second order ODE. The ICs and the equation both suggest that you can consider this equation as one in which the independent variable (t) is missing.


p\left( x \right) = \frac{{dx}}{{dt}} \Rightarrow \frac{{d^2 x}}{{dt^2 }} = \frac{d}{{dt}}\left( {p\left( x \right)} \right) = \frac{{dp}}{{dx}}\frac{{dx}}{{dt}} = p\frac{{dp}}{{dx}}


The DE becomes:


p\frac{{dp}}{{dx}} - 12x = - 28
which is separable after a little rearrangement.

You should be able to solve it from there.

(if you run into problems just reply to this thread - I haven't done the whole question so I don't know if the problem is easy or not)

As for your other question I would interpret it as meaning a = kt^2 .

Edit: Fixed statement above regarding the type of the DE.

[hotvette]

I'm still not sure what they mean in the question with directly proportional.

Benny is right. It just means that a = kt^2 , where k is a constant. The use of the word 'directly' is rather redundant.