practice

  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 − cos (t / 100) vy = 0.0 + 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. In addition, the initial coordinates of the ship were (0,0); that is, the ship started acting this way when it was located at the origin.

    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, and
    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)
       
    Where … 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
       
    Since sine is a circular function, the range of parameter values should be thought of in terms of laps around the unit circle. We will use radians for all angular measures, so be sure your calculator mode or computer preferences are set appropriately.
    1. Let the amplitudes and angular frequencies equal one (A = B = a = b = 1). Set the parameter range to 0 < t < 2π with a reasonable size increment (something that your calculator or computer can complete in under thirty seconds). Draw the lissajous figures for various phase angles. Try simple fractions of a complete circle like φ = 0, ⅙π, ¼π, ½π, ⅔π, 1π, 1½π.
      1. What is the domain of x and y in all cases?
      2. What effect does phase angle have on the lissajous figure?
    2. Let the amplitudes equal one (A = B = 1). Let the phase angle equal a quarter lap around the unit circle (φ = ½π). Draw the lissajous figures for various angular frequencies.You will need to increase the maximum parameter value to 4π, 6π, 20π or higher depending on your choice of frequencies.
      1. Try different whole number values to start with and do not choose the same value for each frequency (a, b ∈ ℤ+ and a ≠ b). Start with small numbers like 1, 2, 3, 4, 5. Try larger numbers if you have the patience. Be sure to vary each frequency. How is the appearance of the lissajous figure affected by your choice of angular frequencies?
      2. Set one of the frequencies to 1 and the other to an irrational number like √2 or π. How is the behavior of this lissajous figure different from those of all your previous trials?
  4. Write something completely different.

numerical

  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 by the Kenner Products toy company in 1965.
    hypocycloid epicycloid
    x =  (A − B) cos t  +  B cos  ⎛
    ⎝    		 A − B  t ⎞
    ⎠
    B
    y =  (A − B) sin t  −  B sin  ⎛
    ⎝    		 A − B  t ⎞
    ⎠
    B
    x =  (A + B) cos t  −  B cos  ⎛
    ⎝    		 A + B  t ⎞
    ⎠
    B
    y =  (A + B) sin t  −  B sin  ⎛
    ⎝    		 A + B  t ⎞
    ⎠
    B
       
  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.
    a.  circle x = a cos t y = a sin t
    b.  ellipse x = a cos t y = b sin t
    c.  cycloid x = at − b sin t y = a − b cos t
    d.  deltoid x = 2a cos t + a cos 2t y = 2a sin t − a sin 2t
    e.  astroid x = a cos3 t y = a sin3 t
    f.  nephroid x = ½ a (3 cos t − cos 3t) y = ½ a (3 sin t − sin 3t)
    g.  folium of descartes x = (3at) / (1 + t3) y = (3at2) / (1 + t3)
    h.  involute of circle x = a cos t + at sin t y = a sin t − at cos t
    i.  serpentine x = a cot t y = b (sin t) (cos t)
    j.  witch of agnesi x = a cot t y = b sin2 t
    k.  pusuit curve, tractrix x = t − a tanh (t / a) y = a sech (t / a)