Question

Particle Motion: A particle moves along the x-axis so that its position at any time t is given by x(t)= t^4-6t^2-3.

Answer 1

Which of the following best describes the motion of the particle for 0<t<1?

A) Moving to the right and speeding up
B) Moving to the left and speeding up
C)Moving to the right and slowing down.
D) Moving to the left at a constant speed
E)Moving to the left and slowing down

Answer 2

first derivative
second derivative
check the signs of each on that interval

Answer 3

So I'll be looking for the velocity and acceleration?

Answer 4

ok :)

Answer 5

yeah exactly
if velocity is positive, forward
if the acceleration is positive, speeding up
etc

Answer 6

guess i meant "moving right"

Answer 7

ok so
v(t)=4t^3-12t
a(t)=12t^2-12

Answer 8

Wait, don't I have to factor ?

Answer 9

find what t equals and then plug in?

Answer 10

you need to factor...

\[x'(t) = 4t(t^2 - 3)\]

set x'(t) = 0 and find the stationary points...

\[x''(t) = 12(t^2 - 1)\]

Answer 11

t= \[\pm \sqrt{1}\]

Answer 12

So the particle is at rest when t=+-sqrt(1)

Answer 13

so the velocity is zero at \[t = 0 ~~~and~~~ t = \sqrt{3}\]

so substitute 0.5 into the velocity and you get a negative value for velocity

I did this because it is on the given interval...

so the slope of the tangent is negative between 0 and 1

so the particle is moving left...

now you can look at the acceleration

Answer 14

well you can't have negative time... so the acceleration is zero at t = 1

and the acceleration is -12 at t = 0

so acceleration is going from -12 at t = 0 to 0 at t = 1

what do you think happened...?

Answer 15

Hmmm so the particle stayed at constant velocity when t=1?

Answer 16

well I think the particle is moving left and slowing.... that's my best guess.

Answer 17

so t= time right? how come the acceleration is -12 at zero seconds (time)?

Answer 18

well the velocity is zero, and by the definitions of velocity and acceleration, this is where the acceleration is a maximum... remember t = 0 is where velocity is 0.

Answer 19

oooooh yeah lol

Answer 20

Well, thanks so much for the help :D
I gotta go now :)

Answer 21

here is a summary that may explain things

https://apmathteacher.wikispaces.com/file/view/Particle+Motion.pdf