Particle's Equation, Velocity and Acceleration

  • #1

Homework Statement


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.
Capture.PNG


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:
[ 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
-----------------------------

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?

thanks in advance
 

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Answers and Replies

  • #2
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,798
1,670

Homework Statement


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.
View attachment 101452

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:
[ 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
-----------------------------

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?

thanks in advance
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
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,798
1,670
  • #4
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


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.
View attachment 101452

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
Got it! Thanks for your guys' help!!
 

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