1. ## Some kinematics/vector calculus

Hi, im having problems solving some questions:

1. The position of a particle is given by $\displaystyle r(t) = (2sin(3t))i + (cos(3t))j$
What is its maximum speed?

PS: what's with the latex system? When will it be back? Should i use standard ASCII for now?

I will post more questions in this thread if need be.

EDIT: sorry, i realized i should post questions in separate threads, so i'll do that instead from now on.

2. Originally Posted by scorpion007
Hi, im having problems solving some questions:

1. The position of a particle is given by $\displaystyle r(t) = (2sin(3t))i + (cos(3t))j$
What is its maximum speed?

PS: what's with the latex system? When will it be back? Should i use standard ASCII for now?

I will post more questions in this thread if need be.

EDIT: sorry, i realized i should post questions in separate threads, so i'll do that instead from now on.
To find the max speed you need to find the time derivative:
v(t) = 6 cos(3t) i - 3 sin(3t) j (This is the velocity, incidentally. You gave the displacement function, a vector.)

What is the speed of this particle?
v(t) = sqrt( v_x ^2 + v_y ^2) = sqrt(36 cos^2(3t) + 9 sin^2(3t))
v(t) = sqrt(27 cos^2(3t) + 9)

So maximum speed will occur when a(t) is 0:
a(t) = (-162 sin(3t) cos(3t)) / (2 [sqrt(36 cos^2(3t) + 9 sin^2(3t))]) = 0

Or when -162 sin(3t) cos(3t) = 0.

This equation has a quite simple solution: t = 0 can be seen by inspection. I leave it to you to show that this is a maximum speed, not a minimum.

v(0) = sqrt(27 + 9) = sqrt(36) = 6 (in whatever units you are using.)

-Dan

3. where did you get this line from?

So maximum speed will occur when a(t) is 0:
a(t) = (-162 sin(3t) cos(3t)) / (2 [sqrt(36 cos^2(3t) + 9 sin^2(3t))]) = 0
the derivative of v(t) does not yield that does it?

4. Originally Posted by scorpion007
where did you get this line from?

the derivative of v(t) does not yield that does it?
Try:

v^2=27 cos^2(3t)+9,

then

2v.a=-162 cos(3t)sin(3t)

a(t)=-162 cos(3t)sin(3t)/v(t),

which when you substitute the expression topsquark gave for v(t)
will give the expression for a(t).

RonL

5. Originally Posted by scorpion007
where did you get this line from?

the derivative of v(t) does not yield that does it?
Sorry. I had made a mistake when I was writing the post out. I guess I didn't correct all of the mistakes before I posted.

-Dan