The Physics
Opus in profectus

Parametric Equations

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  1. An alien is flying her spaceship at half the speed of light in the positive x direction when the autopilot begins accelerating the ship uniformly in the negative y direction at 2.34 m/s2 (0.2 times the acceleration due to gravity on the alien's home planet, the name of which is impossible to write in human symbols). Determine the resultant displacement and velocity of the spacecraft when the acceleration ceases 137 Earth days later.
  2. An alien spacecraft accidentally flies into a plasma cloud (a collection of ionized gas). This disrupts the ship's guidance system, which makes the velocity varying according to the following parametric equations.

    vx = 0.4 − 1.0 cos(t/100)

    vy = 0.0 + 1.0 sin(t/100)

    To make the calculations simpler for us humans, the aliens have adapted this problem to our standards. The equations use m/s for velocity, seconds for time, and radians for angular measure. The initial coordinates of the ship were (0 m, 0 m). Determine…
    1. the displacement as a function of time
    2. the path of the ship for the first 2000 s
    3. the direction of the ship at the beginning and end of this interval
    4. the maximum and minimum speed of the ship
  3. The parametric equations below are used for generating an interesting family of curves called lissajous figures.
     x = A sin (at + φ) 
     y = B sin (bt) 
    xy =  coordinates on a plane
    t =  parameter
    AB =  amplitudes
    ab =  angular frequencies
    φ =  phase angle
    Use a graphing calculator or computer capable of graphing two-dimensional parametric equations. Set the window dimensions to something like
     −1.5 < x < +1.5 
     −1.5 < y < +1.5 
    Let the amplitudes equal one for all problems (A = B = 1). Use radians for all angular measures.
    1. Set the parameter range to one lap around the unit circle (0 < t < 2π).
      1. Set the angular frequencies to one (a = b = 1). Draw the lissajous figures for various phase angles φ (e.g.; 0, π/4, π/2, π, …). What effect does this have on the lissajous figure?
      2. Set the phase angle equal to zero (φ = 0). Draw the lissajous figures for various whole number angular frequencies a and b (e.g.; 1 and 2, 2 and 5, 3 and 8, …). What effect does this have on the lissajous figure?
    2. Set the parameter range to one hundred laps around the unit circle (0 < t < 200π) or some other large number. Leave the phase angle set to zero (φ = 0).
      1. Set one of the frequencies equal to 1 and the other equal to an irrational number like √2, π, or e. What effect does this have on the lissajous figure?


  1. Given the Lissajous curve with the following equations…
    x = sin (2t + π/2) 
    y = sin (t) 
    1. Prove that these parametric equations trace out a parabola on the xy plane.
    2. Locate the points where the curve intersects…
      1. the x axis
      2. the y axis


  1. The parametric equations below are used for generating an interesting family of curves that are informally called spirograph curves in honor of the mechanical drawing toy first manufactured in 1965 by Kenner Products. Here, A and B are adjustable constants and t is the parameter.
    x =  (A − B)cos t  +  B cos 

    A − B  t

    y =  (A − B)sin t  −  B sin 

    A − B  t

    x =  (A + B)cos t  −  B cos 

    A + B  t

    y =  (A + B)sin t  −  B sin 

    A + B  t

    Try drawing some spirograph curves over parameter values of 0 ≤ t ≤ 10π where…
    1. A and B are small whole numbers (like 2, 7, 12, etc.)
    2. A and B are small real numbers that aren't whole numbers (like 3.1, 6.8, 11.75, etc.)
    Start with window dimensions of something like −10 ≤ x, y ≤ +10 and adjust as needed.
  2. Some simple parametric curves to try. The symbols are as follows: x and y are coordinates, a and b are constants, and t is the parameter.
    1. circle
      x = a cos t
      y = a sin t
    2. ellipse
      x = a cos t
      y = b sin t
    3. cycloid
      x = at − b sin t
      y = a − b cos t
    4. deltoid
      x = 2a cos t + a cos 2t
      y = 2a sin t − a sin 2t
    5. astroid
      x = a cos3 t
      y = a sin3 t
    6. nephroid
      x = ½a(3 cos t − cos 3t)
      y = ½a(3 sin t − sin 3t)
    7. folium of descartes
      x = (3at)/(1 + t3)
      y = (3at2)/(1 + t3)
    8. involute of circle
      x = a cos t + at sin t
      y = a sin t − at cos t
    9. sepentine
      x = a cot t
      y = b(sin t)(cos t)
    10. witch of agnesi
      x = a cot t
      y = b sin2 t
    11. pursuit curve
      x = t − a tanh (t/a)
      y = a sech (t/a)
  3. Welcome to the devil's curve. The symbols are as follows: x and y are coordinates, a and b are constants, and t is the parameter.
    x = cos t 

    a2 sin2 t − b2 cos2 t

    sin2 t − cos2 t  
    y = sin t 

    a2 sin2 t − b2 cos2 t

    sin2 t − cos2 t  
    Don't be alarmed. Your calculator won't summon Lucifer from the depths of hell during this exercise. That skill is taught in law school. This curve looks like a toy called a diabolo — a word that sounds very much like diavolo, the Italian word for the devil. Somewhere along the way la curva del diabolo (the diabolo curve) became la curva del diavolo (the devil's curve). A diabolo is a two headed top that is balanced across a string suspended from two sticks — something like a yo yo that can be tossed into the air. The origin of the name may come from the Greek phrase δια βάλλω (dia ballo), which means to "throw across".
    1. Using a graphing calculator or appropriate computer application, graph the devil's curve for several different values of a and b. (Stick with simple values like 1, 2, 3, 4.)
    2. Which combinations of a and b produce a horizontal diabolo at the center of the devil's curve? Which produce a vertical diabolo?
    3. What relationship between a and b determines the orientation of the diabolo?