Solving Parametrics: Sketch, Min Value, & Speed/Accel

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SUMMARY

The discussion focuses on solving parametric equations for a particle's motion in the xy-plane, specifically defined by the equations x(t) = (t²/2) - ln(1+t) and y(t) = 3sin(t) for t in the interval [0, π]. The minimum value of x(t) occurs at t = 0 and t = π, with the corresponding positions being (x(0), y(0)) and (x(π), y(π)). Additionally, the particle is on the y-axis when x(t) = 0, and the speed and acceleration vectors need to be calculated at this point.

PREREQUISITES
  • Understanding of parametric equations
  • Knowledge of calculus, specifically derivatives and optimization
  • Familiarity with trigonometric functions, particularly sine
  • Ability to compute speed and acceleration vectors in parametric motion
NEXT STEPS
  • Calculate the derivative of x(t) to find critical points for minimum value determination
  • Explore the implications of the particle's position on the y-axis by solving x(t) = 0
  • Learn how to derive speed and acceleration vectors from parametric equations
  • Investigate the behavior of the sine function in the context of parametric motion
USEFUL FOR

Students studying calculus, particularly those focusing on parametric equations and motion analysis, as well as educators looking for examples in teaching optimization and vector analysis.

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Homework Statement


A particle moves in the xy-plane so that its position at any time t, 0 less than or equal to t, less than or equal to pi, is given by x(t) = ((t squared)/2))-ln(1+t) and y(t) = 3sint

a. sketch
b. At what time t, 0 less than or equal to t, less than or equal to pi, does x(t) attain its minimum value? What is the position (x(t), y(t)) of the particle at this time?
c. At what time t, 0 less than or equal to t, less than or equal to pi, is the particle on the y-axis? Find the speed and the acceleration vector of the particle at this time.



Homework Equations





The Attempt at a Solution



b. t=0, pi

c. set x(t) function to 0?
 
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c. Yes.
b. Can you describe how you got this answer?
 
Well it seems like you know what to do for the problem but you didn't show us any work for how you got the answer. So we can't help much
 

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