# Math Help - Parametric/velocity/distance Q

1. ## Parametric/velocity/distance Q

A particle moves along a path given by:
x(t) = 3cos(πt) and y(t) = 5sin(πt) and 0≤t≤6

1) Velocity vector fot the particle at any time t?
I just took the derivative of x(t) and y(t). Is that correct?

2) Write an integral expression that gives the distance between t=1.25 and t=1.75
Not sure what to do here.

Help appreciated.

2. Hello, Nerd!

A particle moves along a path given by:
. . x(t) .= .3·cos(πt)
. . y(t) .= .5·sin(πt)
where 0 ≤ t ≤ 6.

1) Velocity vector fot the particle at any time t?
I just took the derivative of x(t) and y(t). Is that correct?
What did you do with the two expressions?
The particle's position vector is: .s .= .[3·cos(πt)]i + [5·sin(πt)]j

Its velocity vector is: .v .= .[-3π·sin(πt)]i + [5π·cos(πt)]j

2) Write an integral expression that gives the distance
between t = 1.25 and t = 1.75
Do they mean "integral" or "integer"? .Either way, it doesn't make sense.

. . . . . . . . . . . . . . . . . . . . . . . . . . . ._ . . . . ._
When t = 5π/4, the particle is at: P(-3√2/2, -5√2/2)
. . . . . . - . . - . . . . . . . . . . . . . . . . . _ . . . . ._
When t = 7π/4, the particle is at: Q(3√2/2, -5√2/2)

. . . - . . - . . . . . . . . . _
The distance PQ is: .3√2

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I think I get it . . . They want the Arc Length.
. . . - . . - . . _______________
. . L . = . ∫ √(dx/dt)² + (dy/dt)² dt

. . . . . . . . . . . . . . . .________________________
So we have: .L .= .∫ √9π²·sin²(πt) + 25π²·cos²(πt) dt

. . evaluated from t = 5/4 to t = 7/4.

3. Its velocity vector is: .v .= .[-3π·sin(πt)]i + [5π·cos(πt)]j
My pre-cal teacher is going to be upset at me (I know I learned it)...but I forgot what the i and j stuff means. Please explain? I wrote it out as two separate function (vy(t) and vx(t) which I didn't think the people who made this problem were looking for)
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Edit: Digging into the dark depths of my brain, I remember, actually. Thanks.
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I think I get it . . . They want the Arc Length.
Awesome. I completely forgot about that stuff.

4. And help evaluating this integral would be really nice.