Question

the velocity of a comet in a certain coordinate system is given by v(t)=(4sin 2t, 4cos 2t, 3) with r(0)=(-2,0,0). Find the distance d of the comet from the origin at time t=1.

I thought to take the integral of v(t), but not sure what else

Answer 1

after you integrate v(t) you get r(t) the position vector
the distance from origin to r(1) is simply
\[d = \sqrt{x^{2} +y^{2} +z^{2}}\]

Answer 2

So am I finding the magnitude after I integrate?

Answer 3

yes after you have components for r(1)

Answer 4

and what are the definite integrals 1 & t? or 1 and 0?

Answer 5

it would be 0 and 1 .... but since they give you intial vector r(0) you can use indefinite integral and solve for constant

Answer 6

ok, can I check back with you to see if I get the answer?

Answer 7

sure

Answer 8

thanks

Answer 9

so for r(1)=(-2cos2, 2sin2, 3) ?

Answer 10

I've substituted 1 for t so I don't have any variables for r(1). Is this correct so far?

Answer 11

yes
now find distance or magnitude of vector

Answer 12

ok, I've got: sqrt[4(cos2)^2 + 4(sin2)^2 +9]. Can (cos2)^2 + (sin2)^2 =1? so it would be sqrt(4*1+9)?

Answer 13

correct

Answer 14

so it's just sqrt(13)

Answer 15

yes

Answer 16

that's the distance?

Answer 17

yes at t=1
so after 1 unit of time the comet is sqrt(13) units away

Answer 18

Sorry for all the questions. I'm doing all this by just trying to think about the ? We didn't learn this in class and it's not in my book, so thank you!

Answer 19

that makes sense!

Answer 20

yw