# Math Help - Particle movement through space #2

1. ## Particle movement through space #2

A particle moves according to a law of motion s = f(t), t >= 0, where t is measured in seconds and s in feet.

Find the velocity at time t.
What is the velocity after 3 s?
When is the particle at rest?
When is the particle moving in the positive direction?
Find the total distance traveled during the first 8 seconds.
Find the acceleration at time t after 3 s.
When is the particle speeding up? When is it slowing down?

The equation is: f(t) = te^(-t/2)

I don't quite understand how do to this problem so a complete explanation would be best.

2. Originally Posted by Neversh
A particle moves according to a law of motion s = f(t), t >= 0, where t is measured in seconds and s in feet.

Find the velocity at time t.
What is the velocity after 3 s?
When is the particle at rest?
When is the particle moving in the positive direction?
Find the total distance traveled during the first 8 seconds.
Find the acceleration at time t after 3 s.
When is the particle speeding up? When is it slowing down?

The equation is: f(t) = te^(-t/2)

I don't quite understand how do to this problem so a complete explanation would be best.
The velocity is:

$v(t)=\frac{ds}{dt}=\frac{d}{dt} \left(t e^{-t/2} \right)$

which should allow you to answer the first two questions.

The third question is asking at what time is $v(t)=0$

The fourth is asking when is $v(t)>0$

I will skip over the fifth

For the sixth you need to know that acceleration:

$a(t)=\frac{d}{dt}v(t)$

The last is asking when is $a(t)>0$

CB

3. Originally Posted by Neversh
A particle moves according to a law of motion s = f(t), t >= 0, where t is measured in seconds and s in feet.

Find the velocity at time t.
What is the velocity after 3 s?
When is the particle at rest?
When is the particle moving in the positive direction?
Find the total distance traveled during the first 8 seconds.
Find the acceleration at time t after 3 s.
When is the particle speeding up? When is it slowing down?

The equation is: f(t) = te^(-t/2)

I don't quite understand how do to this problem so a complete explanation would be best.
For #5, draw a graph of s versus t over the interval t = 0 to t = 8. You will need to calculate the s-coordinate of the turning point (use your answer to #3, which I will denote by $t = \alpha$). The distance travelled should then be easily seen to be $[s(\alpha) - s(0)] + [s(\alpha) - s(8)] = ....$