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Homework Statement
r(t) is the position of a particle in the xyplane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle’s velocity and acceleration vectors at the given value of t.
Homework Equations
First derivative = velocity
(velocity=distance/time)
Second derivative = acceleration
(acceleration=velocity/time)
The Attempt at a Solution
To find the Equation, I first organize it into a set:
[ e^{t}, 2/9 e^{2t} ]
Then I just plug in the value of t (ln3)
[ e^{ln3}, 2/9 e^{2(ln3)} ]
I then reconstruct the original problem with the new values:
r(t) = e^{ln3} i + 2/9 e^{2(ln3)} j
r(ln3) = e^{1.0986} i + 2/9 e^{2.1972} j
then I change the i / j to x/y
r(ln3) = e^{1.0986} x + 2/9 e^{2.1972} y

As for velocity and acceleration, so far I have figured it like this:
Velocity:
[ e^{t}, 2/9 e^{2t} ]
[ te, 4/9 e^{t} ]
Velocity = tei + 4/9e^{t}j
Acceleration:
[ e, 4/9 te]
Acceleration = tei + 4/9 tej
Am I taking the derivative correctly? As far as I know, e remains as e, even after the derivative, right?
thanks in advance
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