jarp0120
  • jarp0120
calculus:rectilinear motion an object is thrown vertically upward from the ground reaches a certain height after 2 sec and returns to the same height on descent, 8 sec later. find its initial velocity and the height in question.
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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jarp0120
  • jarp0120
|dw:1443848025630:dw|
jarp0120
  • jarp0120
is this right?
Miracrown
  • Miracrown
The upward line is 2s, and the total curve at the top is 8s - That seems right :)

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Miracrown
  • Miracrown
We want to use the idea that acceleration due to gravity is constant to help us Do you know if your class prefers feet or meters? We can use either units, but I don't want to do it in a way you haven't seen =/
Miracrown
  • Miracrown
@jarp0120
jarp0120
  • jarp0120
feet
jarp0120
  • jarp0120
and we need to use derivative to find the answer
Miracrown
  • Miracrown
So an important general law all airborne objects follow is this: d^2 y/dt^2 = -32.2 ft/s^2
Miracrown
  • Miracrown
You know the second derivative of height is a constant
jarp0120
  • jarp0120
so that's that acceleration?
Miracrown
  • Miracrown
Lets think mostly about the 8 seconds it was highest up first. What do you think the first derivative of its height was when it was at the very top?
Miracrown
  • Miracrown
The first derivative is velocity. The second derivative is acceleration We know the second derivative is constant
jarp0120
  • jarp0120
oh yes okay
Miracrown
  • Miracrown
Can you answer my question?
Miracrown
  • Miracrown
If youre not sure yet, that's ok too, its just kind of a shortcut if you realize it
jarp0120
  • jarp0120
i don't know
Miracrown
  • Miracrown
That's ok, lets move forward with the problem a bit and readdress it later, when it comes up. So we have this constant second derivative...But that's not very descriptive. Can you use it to get an equation for the first derivative?
jarp0120
  • jarp0120
is the first derivative v=vo-32t?
Miracrown
  • Miracrown
Yes, that's very well put. :)
Miracrown
  • Miracrown
So, do you have any ideas how we might find vo? Its ok if you dont, but if you do they could be useful
jarp0120
  • jarp0120
i don't really know
Miracrown
  • Miracrown
Dont worry, even if you cant see a way yet, we can forge ahead blindly and well find something useful eventually :P
jarp0120
  • jarp0120
thank you
Miracrown
  • Miracrown
So often when unsure the best thing to do in real life is just fight forward. So lets do that. We found an equation for velocity. Can you find one for height? (y, or the 0th derivative)
jarp0120
  • jarp0120
s=vo-16t^2?
jarp0120
  • jarp0120
s= Vot-16t^2?
jarp0120
  • jarp0120
@miracrown?
jarp0120
  • jarp0120
is the length in 2 sec = to the length in 8 sec?
Miracrown
  • Miracrown
@jarp0120 you said before: s= Vot-16t^2? That's technically correct...did you have a d0 at first, and replace it with zero? or an s0, I guess
jarp0120
  • jarp0120
|dw:1443853241921:dw|
Miracrown
  • Miracrown
So we have an equation for s. But we still dont know vo.Maybe we can figure it out though What two times is the ball at the same height?
Miracrown
  • Miracrown
@jarp0120
jarp0120
  • jarp0120
what?
Miracrown
  • Miracrown
Did you follow what I said? If not, its ok.
jarp0120
  • jarp0120
no hahaha
Miracrown
  • Miracrown
Well remember how the problem is framed, It takes 2s to go up to a certain point and then 8 more seconds to go up, and back down to that point. So wouldnt it have the same height at t=2s and t=10s?
jarp0120
  • jarp0120
yeah
Miracrown
  • Miracrown
So how about you plug in 2 and 10 into your position equations and set them equal since we know 2 and 10 should give the same result
jarp0120
  • jarp0120
yes sir
Miracrown
  • Miracrown
It was a little bit of a trick there. So, try out my setting equal idea you should be able to use it to solve vo
jarp0120
  • jarp0120
what do again in the first derivative?
Miracrown
  • Miracrown
You want to do it with the position equation
Miracrown
  • Miracrown
s = vo*t - 16*t^2
Miracrown
  • Miracrown
you want to set s(2) = s(10)
jarp0120
  • jarp0120
s=Vot-16t^2 d=Vo-32t a=-32 right?
Miracrown
  • Miracrown
s v and a, yes
Miracrown
  • Miracrown
v = vo - 32t, I think you meant? @jarp0120
jarp0120
  • jarp0120
substitute the t?
jarp0120
  • jarp0120
yes
Miracrown
  • Miracrown
yes, you want to sub in 2 and 10 respectively for it
jarp0120
  • jarp0120
s(2)=2Vo-64 s(10)=10Vo-1600
Miracrown
  • Miracrown
Mhm
Miracrown
  • Miracrown
since you know those are equal, you can set them equal to solve for vo
jarp0120
  • jarp0120
yay!!!! Vo=192
Miracrown
  • Miracrown
Hooray! lol
Miracrown
  • Miracrown
and that's in units of ft/sec rememeber
Miracrown
  • Miracrown
Now, can you plug that into the d equation and try to use it to maximize the height?
jarp0120
  • jarp0120
yes thank you!!
Miracrown
  • Miracrown
yw :D
jarp0120
  • jarp0120
salamat po!!
jarp0120
  • jarp0120
yes thank you for your help!!
Miracrown
  • Miracrown
No worries. But hand on, It wants you to find the maximum height too doesn't? Alternately, you can maybe guess when the height is maxed If its the same height at t=2 and t=10 When do you think its at the top of its arc?
jarp0120
  • jarp0120
wait
Miracrown
  • Miracrown
tyt
jarp0120
  • jarp0120
6 sec? when v=0 right?
jarp0120
  • jarp0120
:)
Miracrown
  • Miracrown
Yes! Yes! Yes! Yes!
Miracrown
  • Miracrown
But wait there dear
Miracrown
  • Miracrown
That'll be when its at the top you want to find the height then (so plug in to get s(6))
jarp0120
  • jarp0120
when Vo=192?
Miracrown
  • Miracrown
Yupee. Vo is and always will be 192 now
jarp0120
  • jarp0120
and then
Miracrown
  • Miracrown
You just want to find s(6)
jarp0120
  • jarp0120
what will i do find s again? with t=6?
Miracrown
  • Miracrown
Yes
jarp0120
  • jarp0120
576 ft
Miracrown
  • Miracrown
that will give you maximum height, and then youll be done
Miracrown
  • Miracrown
Yes, 576ft is the max height. You got it girl! :)
jarp0120
  • jarp0120
thanks again but there are still many worded problem to be solved in my book hahah
Miracrown
  • Miracrown
Hahaha. #thestruggleisreal
jarp0120
  • jarp0120
#calculusislove hahahah
Miracrown
  • Miracrown
Welp. You have a great night/day. Keep working hard and don't ever give up!
jarp0120
  • jarp0120
i won't thank you

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