what is the velocity of a particle moving along x-axis is given for t0 by V=(32.0t-2.00t^3) m/s where t is in s. What is the acceleration of the particle when it achieves maximum displacement in the positive x-direction?

Question

what is the velocity of a particle moving along x-axis is given for t>0 by V=(32.0t-2.00t^3) m/s where t is in s. What is the acceleration of the particle when it achieves maximum displacement in the positive x-direction?

anonymous · Answer

Hi! I'd suggest breaking the problem into two parts: first find out at what time the displacement is maximised, and then find out what the acceleration is at that time.

anonymous · Answer

Does that help at all?

anonymous · Answer

ehhh  so to find out where displacement is maximized i would have to take the integral of the original quation?

anonymous · Answer

because that would put me in terms of x

anonymous · Answer

but that doesnt necessarily tell me the time intervals to find displacement over that time

anonymous · Answer

I'm not sure what you mean. Right now you have V in terms of t, so if you integrate it you would get x in terms of t. Would you agree?

anonymous · Answer

yes definitly

anonymous · Answer

Great! Now before we actually start doing any integration, tell me this: what would you do once you got x in terms of t?

anonymous · Answer

(We want to find the time at which x is maximised)

anonymous · Answer

that was my next question lol, because that wont tell me when that displacement is maximized

anonymous · Answer

i could pulg in times for t to find when is greatest x

anonymous · Answer

True, you could do it by trial and error. But there's another way: do you know how to maximise a function?

anonymous · Answer

(Using differentiation or integration?)

anonymous · Answer

differentiation

anonymous · Answer

That's right. So what is the condition for x to be maximised?

anonymous · Answer

the time would be minimized

anonymous · Answer

? is that what you mean

anonymous · Answer

for x to be maximized you would need the largest slope

anonymous · Answer

Um, not quite. OK, let's try a different (easier!) approach. Can you draw (roughly) a graph of V(t)?

anonymous · Answer

yes i think i could

anonymous · Answer

Ok, so start off by finding where V=0, and add it to my picture: |dw:1409522875646:dw|

anonymous · Answer

|dw:1409523011285:dw|

anonymous · Answer

Good start! Now there's one more point to find :)

anonymous · Answer

We're solving 32*t - 2*t^3 = 0. Can you factorise this to find the solutions?

anonymous · Answer

Would you like a hint?

anonymous · Answer

yes

anonymous · Answer

i am working on it now

anonymous · Answer

t=0 and .....

anonymous · Answer

t=4

anonymous · Answer

perfect!

anonymous · Answer

so 4 is my maximum point

anonymous · Answer

Let's add that to our drawing. |dw:1409523268341:dw|