# Thread: [SOLVED] derivative (problem solving)

1. ## [SOLVED] derivative (problem solving)

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. Originally Posted by nameck
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..
You're right about parts i and ii. For iii, simply let t = 2 in the original equation.

3. (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. Originally Posted by nameck
(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)?
Your answers for i and iii are correct. For ii try the following:

$\displaystyle 2t e^{-t} - t^2 e^{-t} = 0$

Take out a factor of $\displaystyle e^{-t}$:

$\displaystyle e^{-t} (2t-t^2) = 0$

Now, for this to be true, either $\displaystyle e^{-t} = 0$, or $\displaystyle 2t-t^2 = 0$

One of the properties of the exponential function is that it is NEVER equal to zero, hence the latter must be true:

$\displaystyle 2t-t^2 = 0$

Can you solve this?