Question

@ganeshie8 @phi

Answer 1

I already found the equation for the velocity and position, if that helps.

Answer 2

can you post them?

Answer 3

Yes. I got \(\Large \sin(\pi t)-\frac{1}{2}\) for the velocity

Answer 4

And I got \(\Large -\frac{1}{2}\cos(\pi t)+\frac{1}{\pi}\) for the position. I'm not sure if this one is correct though

Answer 5

Oh, there's another piece of information. It says x(0) = 0 for the position formula.

Answer 6

we don't know the time, so I would do
\[ \int_0^T \sin(\pi t) -\frac{1}{2} \ dt \]

Answer 7

What do you mean?

Answer 8

to get the distance from the origin , integrate velocity from 0 to some time T

Answer 9

your answer for the distance does not look correct, so redo the integral of the velocity

Answer 10

that would be the position formula I got, wouldn't it? So, do I plug in 0 for position and solve for t to find the answer?

Answer 11

Oh, okay.

Answer 12

Can you check my work to find what I did wrong?

Answer 13

@phi u there?

Answer 14

ok, when integrating sin pi t
but -½ dt gives -t/2 term

Answer 15

in other words, you integrate term by term
\[ \sin \pi t - \frac{1}{2} \]

Answer 16

So are you saying I'm just missing a -t/2 term? Where did that come from exactly?

Answer 17

your velocity is sin pi t - ½

Answer 18

yes, I know.

Answer 19

So, when I do the u-substitution to find s(t), that part looks okay, right?

Answer 20

yes
\[ \int \sin(\pi t) - \frac{1}{2} dt \\ = -\frac{\cos(\pi t)}{\pi} - \frac{t}{2} + c \]

Answer 21

but where does the -t/2 part come from?

Answer 22

if you had just
\[ \int \sin(\pi t) \ dt \]
you would get
\[ -\frac{\cos(\pi t)}{\pi} + c\]

Answer 23

but you have
\[ \int \sin(\pi t) - \frac{1}{2} dt \\ \text{ or } \\
\int \sin(\pi t) \ dt- \int \frac{1}{2} dt\]

Answer 24

if you integrate term by term

Answer 25

I get
\[ d = - \frac{\cos\pi t}{\pi} - \frac{t}{2} + \frac{1}{\pi} \]
and we want to find the first t>0 when d =0
I think we have to solve this numerically.
Wolfram gets this
http://www.wolframalpha.com/input/?i=-1%2Fpi+cos%28pi+*+t%29+-+t%2F2+%2B1%2Fpi%3D0