# Example 12.1
FPS()
BEGIN
   t = 0 s
   v = 3 ft / s^3 * t^2 + 2 ft / s^2 * t
   x = 1 ft / s^3 * t^3 + 1 ft / s^2 * t^2
   a = 6 ft / s^3 * t + 2 ft / s^2
END
func(t,3 s)

# Example 12.2
MKS()
t = 4 s
v = 1 m/s
c = -0.4 s / m^2
a = c * v^3
v0 = 60 m/s
vv = 1 / sqrt(1 / v0^2 + -2 * c * t)
ss = -1 / c * (sqrt(1 / v0^2 - 2 * c * t) - 1 / v0)

# Example 12.3
MKS()
t = 0 s
va = 75 m/s
ac = -grav()
sa = 40 m
vb = 0 m/s
sb = sa - (va^2 - vb^2) / (2 * ac)
sc = 0
vc = sqrt(vb^2 + 2 * ac * (sc - sb))

# Example 12.4
MKS()
s0 = 100 mm
s1 = 200 mm
v = 2 / s^2 * sqrt(s1^2 - s0^2)

# Example 12.5
MKS()
BEGIN
t = 1 s
v = 3 m / s^3 * t^2 - 6 m / s^2 * t
ss = 1 m/s^3 * t^3 - 3 m / s^2 * t^2
END
s1 = func(t , 0 s) 
s2 = func(t , 2 s)
s3 = func(t , 3.5 s)
st = (4 + 4 + 6.125) m
dels = 6.125 - 0
vave = dels / 3.5 s

# Example 12.9
FPS()
t = 2 s
x = 8 ft / s * t
y = x^2 / 10 ft
r = sqrt(x^2 + y^2)
vx = 8 ft/s
vy = 2 * x * vx / 10 ft
v = sqrt(vx^2 + vy^2)
qv = atan2(vy,vx); deg
ax = 0
ay = 2 * vx^2 / 10 ft 
a = sqrt(ax^2 + ay^2)
qa = atan2(ay,ax); deg

# Example 12.10
MKS(deg)
t = 0.75
r = [0.5 * sin(2 * t), 0.5 * cos(2 * t), -0.2 * t] m
mag(r)
ur = r / mag(r)
alp = acos(ur[1])
bet = acos(ur[2])
gam = acos(ur[3])
v = [cos(2 * t), -sin(2 * t), -0.2 ] m/s
mag(v)
a = [-2 * sin(2 * t), -2 * cos(2 * t), 0 ] m/s^2
mag(a)

# Example 12.11
MKS(deg)
y = -6 m
t = sqrt(2 * y / (-grav()))
v0 = 12 m/s
r = v0 * t

# Example 12.12
FPS(deg)
v0 = 25 ft/s
q = 30 deg
v0x = v0 * cos(q)
v0y = v0 * sin(q)
t = 20 ft / v0x
ya = v0y * t - 0.5 * grav() * t^2

# Example 12.13
MKS(deg)
sb = -1 m
sa = 0 m
q = 30 deg
t = 1.5 s
BEGIN
va = 1 m/s
e = sa + va * t * sin(q) - 0.5 * grav() * t^2 - sb
END
solve(e, va)
r = va * cos(q) * t
BEGIN
h = 0 m
e = va^2 * sin(q)^2 - 2 * grav() * (h - 1 m)
END
solve(e,h)

# Example 12.14
MKS(deg)
v0 = 6 m/s
a = 2 m/s^2
x = 1 / 10 m
y = x^2 / 20 m
vx = x / 10 m
q = 45 deg
va = 6
rho = (abs(1 + va^2))^1/5 / abs(0.1)

# Example 12.15
FPS(deg)
at = 7 ft/s^2
r = 300 ft
BEGIN
t = 1 s
v = at * t
an = v^2 / r
a = sqrt(at^2 + an^2)
e = a - 8 ft /s^2
END
solve(e,t)

# Example 12.16
MKS(deg)
c = (0.2 m/s^2) / (1 s)  # calibrate
t = 1 s
a = c * t
v = (c / 2) * t^2
ss = (c / 6) * t^3
sb = (3 + 2 * pi() * 2 / 4) m
t = (sb * 6 / c)^(1/3)
a = c * t
v = (c / 2) * t^2
r = 2 m
an = v^2 / r
aa = sqrt(a^2 + an^2)

# Example 12.18
MKS( mm, mm/s, mm/s^2)
cq = (1 rad) / (1 s)^3 # calibrate
cr = (100 mm) / (1 s)^2 # calibrate
t = 1 s
q = cq * t^3
qdot = 3 * cq * t^2
qdotdot = 6 * cq * t
r = cr * t^2
rdot = 2 * cr * t
rdotdot = 2 * cr
v = [rdot, r * qdot]
mag(v)
PolarAngle(v); deg
a = [rdotdot - r * qdot^2, r * qdotdot + 2 * rdot * qdot]
mag(a)
PolarAngle(a); deg

# Example 12.19
MKS( )
c = 100 m 
q = 45 deg
qdot = 4 rad /s 
qdotdot = 0 rad /s^2
r = c / cos(q)
rdot = c / cos(q) * tan(q) * qdot
rdotdot = c / cos(q) * tan(q)^2 * qdot^2 + c / (cos(q))^3 * qdot^2
v = [rdot, r * qdot]
mag(v)
PolarAngle(v); deg
a = [rdotdot - r * qdot^2, r * qdotdot + 2 * rdot * qdot]
mag(a)
PolarAngle(a); deg

# Example 12.20
FPS
c = 0.5 ft
v0 = 4 ft/s
a0 = 30 ft/s^2
q = 180 deg
BEGIN
  qdot= 4 rad/s
  qdotdot = 0 rad/s^2
  r = c * (1 - cos(q))
  rdot = c * sin(q) * qdot
  rdotdot = c * cos(q) * qdot^2 + c * sin(q) * qdotdot
  v = [rdot, r * qdot]
  mag(v)
  PolarAngle(v); deg
  a = [rdotdot - r * qdot^2, r * qdotdot + 2 * rdot * qdot]
  mag(a)
  PolarAngle(a); deg
  err1 = mag(a) - a0
  err2 = mag(v) - v0
END
solve(err1,err2, qdot, qdotdot)

# Example 12.21
FPS
vb = -6 ft/s
va = -3 * vb

# Example 12.22
FPS
vb = -6 ft/s
va = -4 * vb

# Example 12.23
MKS
va = 2 m/s
vb = -va / 4

# Example 12.24
MKS(m, mm/s, mm/s^2)
BEGIN
   x = 1 m
   y = sqrt(225 m^2 + x^2) - 15 m
   err = y - 10 m
END
solve(err, x)
vs = 0.5 * 20 (m^2 / s) / sqrt(225 m^2 + x^2)
va = 0.5 m/s
a = 225 m^2 * va^2 / (225 m^2 + x^2)^1.5

# Example 12.25
MKS(mph, deg)
q = 45 deg
va = 45 mph * [ cos(q), sin(q)]
vt[1] = 60 mph
vta = vt - va
mag(vta)
PolarAngle(vta)

# Example 12.26
MKS(kph, km/hr^2, deg)
vb = 600 kph
va = 700 kph
vba = vb - va
r = 400 km
ab0 = vb^2 / r
ab = [ab0, -100 km/hr^2]
aa[2] = 50 km/hr^2
aba = ab - aa
mag(aba)
PolarAngle(aba)

# Example 12.27
MKS(deg)
vb[2] = -10 m/s
q = 60 deg
va = -18 m/s * [cos(q), sin(q)]
vba = vb - va
mag(vba)
PolarAngle(vba)
r = 100 m
abn0 = (12 m/s)^2 / r
ab = [-abn0, -3 m/s^2]
aa = 2 m/s^2 * [cos(q), sin(q)]
aba = ab - aa
mag(aba)
PolarAngle(aba)

# Homework Problems:

TBD