Question

The velocity of a particle moving along the x-axis is v(t) = t2 – 2t, with t measured in minutes and v(t) measured in feet per minute. To the nearest foot find the total distance travelled by the particle from t = 0 to t = 3 minutes.

Answer 1

@zepdrix pretty sure I just integrate, just wanted to confirm

Answer 2

Ummm

Answer 3

https://gyazo.com/1ad2f2f534c3e4c134c1e809420a3e7a

Answer 4

yes integrate between t = 0 and t = 3

Answer 5

I mean, yes, integrating will get you your position function, good.
The issue is,
if the particle moved backwards and then also forwards during those first 3 minutes,
then we might have an issue.

The s(t) is going to tell us about displacement, not total distance.
So maybe we need to look at critical points first.

Answer 6

Maybe I'm mistaken hehe, rusty

Answer 7

It says move along so I doubt it means back and forth.

Answer 8

Anyways I got 0.

Answer 9

yes it does move in 2 directions
0 is the displacement no the distance travelled

Answer 10

Setting v(t)=0 gives us:\[\large\rm 0=t^2-2t\qquad\to\qquad t=0,~t=2\]
These are the times when the velocity changes from positive to negative.
If you do a sign chart or something similar, you see that the particle was moving backwards from t=0 to t=2.

So maybe what you need to do is break up the integral.

Answer 11

yes

Answer 12

\[\large\rm total~distance=\left|\int\limits_0^2 t^2-2t~dt\right|+\int_2^3 t^2-2t~dt\]
Somethingggg like that.

Answer 13

yes

Answer 14

Maybe that's more complicated :)
Just integrate,\[\large\rm s(t)=\frac{1}{3}t^3-t^2+c\]
The particle is moving backwards from t=0 to t=2,
so calculate the distance it covered in that interval,\[\large\rm s(2)-s(0)\]And then calculate the distance it traveled while moving forward,\[\large\rm s(3)-s(2)\]and add those quantities together.

Same thing, just without the ugly integrals I guess.

Answer 15

Nah I like the first one better, actually how I remember doing it.
I got 2.667

Answer 16

the graph looks something like below so you need to calculate the 2 areas