Tried to find the acceleration by doing ma=mg-mkv^2

[XtremePhysX]

Homework Statement

http://i.imgur.com/kK7xE.jpg

Homework Equations

In the picture.

The Attempt at a Solution

I tried to find the acceleration by doing ma=mg-mkv^2.

 

[voko]

I tried to find the acceleration by doing ma=mg-mkv^2.

I suggest that you change the direction of x, to be consistent with the questions.

Now, with the equation of the acceleration and the forces, can you answer (i)?

 

[XtremePhysX]

I suggest that you change the direction of x, to be consistent with the questions.

Now, with the equation of the acceleration and the forces, can you answer (i)?

Can you explain to me the concept of terminal velocity please ?

 

[dydxforsn]

I tried to find the acceleration by doing ma=mg-mkv^2.

That's the right Newton's 2^nd law for the ascension (who knew how to spell this word?..) From that you can parts (ii) and (iii) (hint for part (ii): What happens at the terminal velocity that would allow you to solve for it from that equation?)

For part (iv) you are going to need to do something to the left-hand side of the equation:$$\frac{dv}{dt} = -\frac{g}{{V_t}^{2}} ({V_t}^2 + v^2)$$ Note: For the love of god change upper-case 'V' to '$V_t$' to avoid confusion with lower-case 'v'...
What are you going to have to do to the left-hand side? Well you want just v's and x's, t's are bad because you want some final form x(v). A rule from standard calculus will allow you to write this side in terms of just 'v' and 'x'. Hint: It's a rule so divisive that Joe Biden was in the news for even mentioning it recently.

Edit #1: I notice your mentioning of terminal velocity and perhaps some misunderstanding regarding it. Terminal velocity is the velocity at which a particle is going so fast (due to a velocity independent accelerating force) that some resistive force (that's a function of 'v') will get so large as to perfectly oppose it. In other words, it's a velocity at which there is zero acceleration on the particle. Notice that they make you calculate it on the descent first in part (ii) before part (iii). You need to use the same value you get in part (ii) for part (iii), it's not physically going to change during ascent/descent, but it makes more sense to calculate it for the descent equation because in that equation it's actually something reachable...

Edit #2: Further hint for the left-hand side stuff I was mentioning. You are going to want some form on the left-hand side that contains $\frac{dx}{dt}$ which you are simply going to call 'v'. That's how you get rid of the 't' after you apply the infamous rule I hinted at.

Edit #3: Woops, I put that the left-hand side stuff was for part (iii), I meant to say that it's for part (iv)..

 

[dydxforsn]

First notice the edits to my last post

For part (v) you're going to want to forget about the solution x(v) you found for part (iv) and go back to the equation: $$\frac{dv}{dt} = -\frac{g}{{V_0}^2} ({V_0}^2 + v^2)$$ and actually solve for v(t). Here you want to use '$\frac{{V_o}}{2}$' and 0 as your limits of integration for your velocity integral, you can then solve for your upper limit of your time integral which will simply be the time in question.

Part (vi) is really similar to part (iii).

And finally for part (vii) you are going to want to go back to your solution x(v) and find x(0.1$V_0$). Well... that would be for ascension, I think it wants it for the descent in which case you will have to derive part (iv) for the descent and solve for x(0.1$V_o$) from that equation.. Ugh.

 

[voko]

Can you explain to me the concept of terminal velocity please ?

Observe that the air resistance (drag) grows as the square of the velocity. Which means that at a certain velocity the drag will be equal to the pull of gravity, the net acceleration will be zero, and the velocity will stop growing. This is the terminal velocity.

 

[XtremePhysX]

So how do I do part (i) ?

 

[voko]

"Which means that at a certain velocity the drag will be equal to the pull of gravity, the net acceleration will be zero, and the velocity will stop growing." What does that mean with regard to the equation of motion?

 

[XtremePhysX]

"Which means that at a certain velocity the drag will be equal to the pull of gravity, the net acceleration will be zero, and the velocity will stop growing." What does that mean with regard to the equation of motion?

$$mkV_{t}^{2}=mg$$ therefore $$kV_{t}^{2}=g$$ ?

 

[voko]

Correct.

 

[XtremePhysX]

Correct.

I think once I get part the second part, which is the acceleration, the part after that is just changing the acceleration to vdv/dx and integrating.
But how do I get (iii) ? the acceleration?

 

[voko]

Open the parentheses in (iii), what do you get?

 

[XtremePhysX]

$$v\frac{dv}{dx}=-g-g\frac{v^{2}}{V_{t}^{2}}$$

 

[voko]

How did you get the term on the left? The rightmost term, apply the identity you derived for (ii).