Question

My teacher did one example in class. I'll do a problem, and anyone is invited....If I'm wrong reply or message, but please don't do my prob. for me.

Answer 1

I'll be posting part by part....

Answer 2

I'll be posting part by part....

Answer 3

The position of a particle is given by the equation \(\normalsize\color{blue}{ s=f(t)=t^3-6t^2+9t }\)
where t is measured in seconds and s in meters.

a) Find the velocity at time t.
b) What is the velocity after 2 s? After 4 s?
c) When is the particle at rest?
d) When is the particle moving forward (that is in the positive direction) ?
e) Draw a diagram to represent the motion of the particle.
f) Find the total distance traveled by the particle during the 1st 5 seconds.
g) Find the acceleration at time t and after 4 s.

Answer 4

for part a, the velocity function, is the derivative of the position function.
\(\normalsize\color{blue}{ v(t)=3t^2-12t+9 }\)

( v(t) is the new velocity function)

Answer 5

for part b,
the velocity after 2 s, means the instantaneous velocity, when t=2.
So,
\(\normalsize\color{blue}{ v(2)=3(2)^2-12(2)+9=12-24+9=-3 }\)
(This is ds/dt at t=2.)
SO after 2 s, it is \(\normalsize\color{blue}{ -3~m/s }\)

The velocity after 4 s, is ds/dt at t=4, so
\(\normalsize\color{blue}{ v(4)=3(4)^2-12(4)+9=48-48+9=9 }\)
SO after 4 s, it is \(\normalsize\color{blue}{ 9~m/s }\).

Answer 6

For part c,
we know that the particle is at rest when t=0.
so we set the velocity function equal to zero.

\(\normalsize\color{blue}{ v(t)=3t^2-12t+9=0 }\)
Solving for t,
\(\normalsize\color{blue}{ 3t^2-12t+9=0 }\)
\(\normalsize\color{blue}{ 3(t^2-4t+3)=0 }\)
\(\normalsize\color{blue}{ 3(t-1)(t-3)=0 }\)
So the particle is at rest after 3 s, and 1s.

Answer 7

Part d.
The particle is moving in a positive direction when \(\normalsize\color{blue}{ v(t)>0}\) , and that is: \(\normalsize\color{blue}{ 3(t-1)(t-3)>0 }\) .

the inequality is true when both factors are positive or when both are negative, and thus

\(\normalsize\color{blue}{ t<1 }\) to make both factors negative.
and
\(\normalsize\color{blue}{ t>3 }\) to make both factors positive.

Answer 8

For part e. schematic sketch of the motion of the particle.



this is the motion back and forth along the s-axis.
(s plays the role of the x)