Vector Calculus: Find Position & Velocity Vectors

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Homework Help Overview

The discussion revolves around finding the position and velocity vectors of a particle given its acceleration vector, initial velocity, and initial position. The acceleration is defined as a(t) = −cos t i − sin t j, with initial conditions v(0) = i and r(0) = j.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss integrating the acceleration vector to find the velocity vector and subsequently the position vector. There are attempts to apply initial conditions to determine constants of integration. Some participants express confusion regarding the correct form of the velocity and position vectors, particularly in relation to the initial conditions.

Discussion Status

There is ongoing exploration of the correct expressions for the velocity and position vectors. Participants are questioning the integration steps and the application of initial conditions, with some expressing uncertainty about the vector constants involved. Multiple interpretations of the results are being considered, and guidance has been offered regarding the nature of vector constants.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the information they can use or the methods they can apply. There is a noted discrepancy between their results and those provided in the textbook, leading to further questioning of their reasoning and calculations.

danny_manny
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Homework Statement



Use the given information to find the position and velocity
vectors of the particle.

a(t) = −cos t i − sin t j; v(0) = i; r(0) = j

Homework Equations





The Attempt at a Solution


Ok first step integrate a(t).

which i get to be
-sin(t)i +cos(t)j + c

now using the initial condition v(0) = i
i get v(t) = 1-sin(t)i +cos(t)j

now integrate v(t)

= (cos(t)+t)i +sin(t)j
using initial condition r(0) = j
r(t) = (cos(t) + t)i + (sin(t)+1)j

they are my two answers in bold but the back of my book has different answers and i don't know what's going wrong.

Thanks for the assistance
 
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danny_manny said:

Homework Statement



Use the given information to find the position and velocity
vectors of the particle.

a(t) = −cos t i − sin t j; v(0) = i; r(0) = j

Homework Equations



The Attempt at a Solution


Ok first step integrate a(t).

which i get to be
-sin(t)i +cos(t)j + c

now using the initial condition v(0) = i
i get v(t) = 1-sin(t)i +cos(t)j

now integrate v(t)

= (cos(t)+t)i +sin(t)j
using initial condition r(0) = j
r(t) = (cos(t) + t)i + (sin(t)+1)j

they are my two answers in bold but the back of my book has different answers and i don't know what's going wrong.

Thanks for the assistance
Your v(t) is incorrect. It gives you that v(0) = 1 + j. That's neither a vector nor a scalar. You can't add a vector & a scalar.

The constant, c, you have after integrating a(t) is a vector constant, c.
 
SammyS said:
Your v(t) is incorrect. It gives you that v(0) = 1 + j. That's neither a vector nor a scalar. You can't add a vector & a scalar.

The constant, c, you have after integrating a(t) is a vector constant, c.

so the vector constant should be equal to i no?
 
danny_manny said:
so the vector constant should be equal to i no?
No, although I see that you may have meant to have v(t) = (1-sin(t))i + cos(t)j. But that still ignores the fact that cos(0) = 1, not zero.
 
so the answer must be
(1-sin(t))i+(cos(t)+1)j ?

in the book however the answer is given
(1-sin(t))i+(cos(t)-1)j

I don't know where I'm going wrong.
Thanks again.
 
danny_manny said:
so the answer must be
(1-sin(t))i+(cos(t)+1)j ?

in the book however the answer is given
(1-sin(t))i+(cos(t)-1)j

I don't know where I'm going wrong.
Thanks again.

cos(0) - 1 = 1 - 1 =0
 
Sorry Sammy I'm still lost :(
 
danny_manny said:
so the answer must be
(1-sin(t))i+(cos(t)+1)j ?

in the book however the answer is given
(1-sin(t))i+(cos(t)-1)j
How did you come up with your answer? What do you get when you set t=0 in your answer?
 
v(0) = -sin(0)i +cos(0)t + c
= 0i+1j+c
 
  • #10
I don't see how you solved for c.
 
  • #11
That's the part I'm stuck at do i set the lhs equal to zero?
 
  • #12
What do the initial conditions given in the problem statement say?
 
  • #13
oh I see, thanks.
 

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