# Particle's Equation, Velocity and Acceleration

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1. May 30, 2016

1. The problem statement, all variables and given/known data
r(t) is the position of a particle in the xy-plane 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.

2. Relevant equations
First derivative = velocity
(velocity=distance/time)
Second derivative = acceleration
(acceleration=velocity/time)

3. The attempt at a solution
To find the Equation, I first organize it into a set:
[ et, 2/9 e2t ]
Then I just plug in the value of t (ln3)
[ eln3, 2/9 e2(ln3) ]
I then reconstruct the original problem with the new values:
r(t) = eln3 i + 2/9 e2(ln3) j
r(ln3) = e1.0986 i + 2/9 e2.1972 j
then I change the i / j to x/y
r(ln3) = e1.0986 x + 2/9 e2.1972 y
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As for velocity and acceleration, so far I have figured it like this:
Velocity:
[ et, 2/9 e2t ]
[ te, 4/9 et ]
Velocity = tei + 4/9etj

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?

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2. May 30, 2016

### SteamKing

Staff Emeritus
For the derivatives of et, you should review the rules of differentiation for such functions and not neglect application of the chain rule.

http://www.themathpage.com/acalc/exponential.htm

Remember, et is not differentiated like xn.

3. May 31, 2016

### SteamKing

Staff Emeritus
4. May 31, 2016

### Ray Vickson

You have not done the second part, which says "Find an equation in x and y whose graph is the path of the particle". This means that instead of representing the particle's orbit as $(x(t),y(t))$ you should represent the curve of the orbit (not including "time" information) as an equation of the form $y = F(x)$ or $x = G(y)$ or $H(x,y) = 0$, and you are to figure out the functions $F$ or $G$ or $H$, as needed.

5. May 31, 2016