Question

Please check my work:)

Suppose a particle moves back and forth along a straight line with velocity v(t), measured in feet per second and acceleration a(t).

Answer 1

a) what is the meaning of \[\int\limits_{60}^{120}v(t)dt\]
b) what is the meaning of \[\int\limits_{60}^{120}\left| v(t) \right|dt\]
c) what is the meaning of \[\int\limits_{60}^{120}a(t)dt\]

Answer 2

a) net change (how can I say this better..?)
b) total distance traveled/total change in the interval 60 and 120
c) I'm not sure xD

Answer 3

@zepdrix :)

Answer 4

Hmm I like part b :d

Answer 5

for part a,
maybe we use the word `displacement`.

The integral tells us far away the particle is from it's position at t=60 to t=120.
Somethinggggg like that...
thinking...

Answer 6

hm hm "displacement of the particle" then?

Answer 7

The integral in A tells us how far the particle has been displaced from its position at t=60 to its position at t=120.

The integral in B tells us the total distance the particle traveled from its position at t=60 to its position at t=120.

Answer 8

Ya, whatever wording makes sense to you :)

Answer 9

hehe your wording is much more eloquent.

Answer 10

\[\large\rm \int\limits_{60}^{120}a(t)dt\quad=v(120)-v(60)\]So we still trying to make sense of this last part? :)

Answer 11

mhm.

Answer 12

Hmm I guess we can't use the word displacement this time ;[
So maybe we go back to your original idea:
Net change in velocity from t=60 to t=120.

Answer 13

What does that mean exactly?

Answer 14

I don't think a(t) = net change? o.o

Answer 15

Let's say at t=60,
the particle was moving 50mph.

Then I guess at maybe t=70, it slows down to 30mph.
And then how bout at t=90, it speeds up to 80mph.
And then at t=120, it slows down to a velocity of 75mph.

So the net change in velocity from t=60 to t=120 is the change from the start velocity to the end velocity: 75 - 50 = 25

From 50 to 30 is -20
from 30 to 80 is +50
from 80 to 75 is -5

-20 + 50 - 5 = 25

Answer 16

That was a stupid explanation ^
But read it anyway :P
See if it helps.

Answer 17

hrrm I am beginning to seee xD

Answer 18

We can agree on this, yes?\[\large\rm \int\limits a(t)dt=v(t)\]So then if we give this integral limits we have,\[\large\rm \int\limits_{60}^{120}a(t)dt=v(120)-v(60)\]

Answer 19

Yes :) but it should follow in the opposite direction, ya?\[\large\rm if\quad a(t)=v'(t),\qquad then\quad \int\limits a(t)=\int\limits v'(t)=v(t)\]

Answer 20

ou, yes :>

Answer 21

forgot about the integral being in front of it..haha..ha.

Answer 22

Yes, I agree!