Question

A particle travels on the x axis so its velocity at time t is given by v(t)=cost-(1/2) for 0<=t<=2pi. The particle is at position x=2 at time t=0.
I need help with how to find:
For what values of t is the particle moving to the left? What is the particles final position?

Answer 1

\(v(t) = \cos(t) - 1/2\)?

\(v(t) = \cos(t - 1/2)\)?

\(v(t) = \cos(1/\sqrt{t})\)?

You DO have to write clearly.

Answer 2

sorry
\[v(t)=\cos(t)-\frac{ 1 }{ 2 }\]

Answer 3

Are you aware of the relationship between the location or displacement function and the anti-derivative of the velocity function?

If x(t) is the displacement function, then \(\dfrac{d}{dx}x(t) = v(t)\)

Ringing any bells?

Answer 4

yes.

Answer 5

Okay, very good. Then we have v(t) and we need to find x(t). How do we do that?

Answer 6

v(t)=x'(t)

Answer 7

Whoops! I said dx(t)/dx up above. I meant dx(t)/dt. Sorry about that.

Okay, v(t) = x'(t). We have an explicit expression for v(t), how do we get an explicit expression for x(t)?

Answer 8

do the integral?

Answer 9

Okay, let's see it. Don't forget the "+ C" part. It's very important.

Answer 10

particle moves left when v(t) < 0

Answer 11

wait so i do the integral of v(t)?

Answer 12

Why do you still doubt? Do it!

Answer 13

No, you don't integrate. The position of the particle at a given time does not tell you whether it goes left or right. It only tells you where the particle is at.

To find which direction it goes, you will need to compare its velocity with time.
it goes left if v(t) < 0 and it goes right if v(t) > 0

Answer 14

:( so do i plug in values of t? by the way, this is a question with a graphing calculator.

Answer 15

you to solve cos(t) - 1/2 < 0
which gives you pi/3 < t < 5pi/3

http://www.wolframalpha.com/input/?i=solve+cos%28t%29+-+1%2F2+%3C+0+from+0+to+2pi