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:
[tex] \frac{d}{dx} f(g(x)) = f\;'(g(x)) \cdot g'(x) [/tex]
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:
[tex] \int_{v_o}^{v_1} \frac{dv}{v} [/tex]
[tex] \int_{x_o}^x \frac{dy}{(y + x_o)^2} [/tex]
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

 

[rwisz]

for reaching out! I am happy to assist with your questions on introductory mechanics. It's great that you have a friend who suggested this board for help. I understand the importance of seeking help when needed.

For question 14, the first step would be to use the chain rule to find the derivative of each term separately. For part (a), the derivative of A cos (at) would be -A sin (at) and the derivative of -gt^2/2 would be -gt. Then, we would use the sum rule to combine the two derivatives to get the final answer of -A sin (at) - gt. For part (b), we would use the product rule to find the derivative of B1 exp(-yt) and B2t exp(-yt), and then use the sum rule to combine them.

For question 15, we would use the substitution method to solve the integrals. For part (a), let v = v1. Then, dv = 0 and v1 = v. The integral would become the natural log of v, evaluated from vo to v1. For part (b), let y = y + xo. Then, dy = dx and y = x. The integral would then become x/(y+xo), evaluated from xo to x.

I hope this helps and good luck on your assignment! Remember, it's always important to seek help when needed and to not be afraid to ask questions. Keep up the hard work!