Hey, I'd love a hand - Introductory Mechanics

[Gordanier]

Hi everybody, thank you in advance for all comments/help.

First off, I know I am new here, but a friend of mine is a casual on this board and he suggested I try it for help.

I have just begun a 2nd year university introductory to mechanics class, and I have an assignment due tomorrow. I know youre all thinking I am a slacker and put it all off, but I finished 90% of it, just stuck on the last 2 questions that me and a co-student have been going over notoriously on a white board for the last couple hours.

They are derivative/integral questions with the following information:

"14. Calculate the derivative df/dt, where
(a) f(t) = A cos (at - gt^2 /2)
(b) f(t) = B1 exp(-yt) + B2t exp (-yt). (i think exp means exponent, and the y is latin gamma?

15. Calculate the following integrals:
(a) v1 dv/v (v1 > vo> 0)

vo

(b) x dy / (y+xo)^2 (x > xo > 0)

xo


I know it says an attempt at a solution, but everything we've attempted so far has been on a whiteboard and i don't think I am getting anywhere.

I would very much appreciate any help whatsoever as I am taking this class as an elective since I am interested in the field, but have exhausted my resources for these questions and do not know where else to turn.

Thanks

 

[gneill]

"14. Calculate the derivative df/dt, where
(a) f(t) = A cos (at - gt^2 /2)

You should know the calculus rule for taking the derivative of 'nested' functions:
$$\frac{d}{dx} f(g(x)) = f\;'(g(x)) \cdot g'(x)$$
Here f() is cos() , g() is at - gt²/2 , and the variable you're differentiating with respect to is t.

(b) f(t) = B1 exp(-yt) + B2t exp (-yt). (i think exp means exponent, and the y is latin gamma?

exp(x) usually designates the exponential function, e^x.

15. Calculate the following integrals:
(a) v1 dv/v (v1 > vo> 0)

vo

(b) x dy / (y+xo)^2 (x > xo > 0)

xo

Presumably those are meant to be:
$$\int_{v_o}^{v_1} \frac{dv}{v}$$
$$\int_{x_o}^x \frac{dy}{(y + x_o)^2}$$
The above are common integrals that you should be able to find in a table of integrals (particularly (a), which is very common indeed). You should have the indefinite integral for (a) memorized, since it's so common. Finding the definite integral is just a matter of applying the integration limits to it.

(b) can be solved with an appropriate change of variables to cast it in the form dz/z², which is another very common integral.

 

[Gordanier]

Yes you are correct with the proper notations, thank you for that. Also, thanks for taking the time to answer.

I'm sure tonight would have gone much easier if I had the textbook, but I opted to pay $40 on ebay instead of $200 in the bookstore, and it hasn't arrived yet :) So that is why I am in dire straights, but if the equation is as common as you say then I'm sorry for asking a dumb question :)

Also thank you for the differentiation help, as with the integrals, wow. I was way over thinking them. Thank you very much gneill, saved my GPA! (not really, assignment was only worth 2% and got most of them already, but nonetheless, thanks).

 

[gneill]

If you get stuck with integrations, or stuck without your table of integrals or crib sheet, often the Wolfram Online Integrator can help. Google will find it for you