Derivative (problem solving)

1. Dec 11, 2009

naspek

Let s = s(t) be the position function of a particle moving in a straight line.
Suppose that the position of the particle is given by the formula
s (t) = t^2 e^-t ; t >= 0
where t is measured in seconds and s in meters.

(i) Find the velocity of the particle at time t.
* should i differentiate s(t) to get the velocity?

(ii) When is the particle at rest?
* when velocity = 0.. am i right?

(iii) Find the total distance traveled by the particle during the first two seconds.
*dont have any idea..

2. Dec 11, 2009

Dick

3. Dec 12, 2009

naspek

(i) Find the velocity of the particle at time t.
* should i differentiate s(t) to get the velocity?

answer--> (2t e^-t) - (t^2 e^-t)

(ii) When is the particle at rest?
* when velocity = 0.. am i right?

(2t e^-t) - (t^2 e^-t) = 0
(2t e^-t) = (t^2 e^-t)
(2e^-t) = (t e^-t)
*how to solve t?

(iii) Find the total distance traveled by the particle during the first two seconds.
*dont have any idea..

at t = 2
s (t) = t^2 e^-t
s (2) = 2^2 e^-2
.......= 0.54 meters

is my answer for (i) and (iii) is correct?
how am i going to solve (ii)?

4. Dec 12, 2009

mg0stisha

Are you just trying to solve (ii) for t?

5. Dec 12, 2009

naspek

yes. because i need the value of 't' when the particle at rest..

6. Dec 12, 2009

mg0stisha

do you see anything you can factor out from both sides?

7. Dec 12, 2009

naspek

(2t e^-t) - (t^2 e^-t) = 0
(2t e^-t) = (t^2 e^-t)
(2e^-t) = (t e^-t)
e^-t(2) = e^-t (t)
(e^-t)/(e^-t) 2 = t

*(e^-t)/(e^-t) = 1?

8. Dec 12, 2009

Staff: Mentor

You should write an equation; namely v(t) = 2te^(-t) - t^2*e^(-t)
Yes.
I'm going to cut in here because the rest of your work doesn't help you get where you need to go. It's not wrong, but it isn't helpful either.
s'(t) = v(t) = 2t e^(-t) - t^2 e^(-t) = 0
v(t) = 0 ==> 2t e^(-t) - t^2 e^(-t) = 0 = 0 ==> (2t - t^2)e^(-t) = 0
e^(-t) is never 0. When is 2t - t^2 = 0? Those are the times when v(t) = 0.

Integrate the velocity between t = 0 and t = 2.

9. Dec 12, 2009

naspek

for (iii) 4e^-2
am i got it right?

Last edited: Dec 12, 2009
10. Dec 12, 2009

HallsofIvy

Yes, of course. You were told in the problem itself that s(t)= t2e-t. The answere to (iii) is just s(2).

11. Dec 12, 2009

naspek

Thank u for the confirmation of my answer.. =)