3.1.K.sagews

import matplotlib.pyplot as plt
from numpy import linspace, full_like, array, arange
# 3
from mpmath import degrees, radians

# 1
from sympy import *
from sympy import N as Num

# 4
# pie is better than pi
pie = 2*pi

# 5
# SI units:
m  = Symbol("m",  positive=True)
s  = Symbol("s",  positive=True)
kg = Symbol("kg", positive=True)

print("\n--- User input -----------------------")

symbolic = True
symbolic = False

if symbolic:
    # quantity = symbol:
    alpha= var("alpha")
    g = var("g")
    h, xP = var("h, x_P")
else:
    # variable = quantity:
    # quantity =  factor times unit:
    # pie/12 = 30 deg
    # pie = 2 pi
    alpha = pie/12
    g = 10 *m/s/s
    h = 1 *m
    xP = 3 *m

print("\n--- a: -------------------------------")

sa = sin(alpha)
ca = cos(alpha)
ta = tan(alpha)
ss = sa*sa
cc = ca*ca
s2a = sin(2*alpha)

[pprint(latex(x)) for x in [sa,ss,ca,cc,ta,s2a]]

x = var("x")
v0 = var("v0", positive=True)

z1 = -g/2/v0/v0/cc *x*x + ta*x

pprint(latex(z1))

print("\n--- b: -------------------------------")

rhs = z1.subs(x, 3*m)
eq = Eq(-h, rhs)

sol = solve(eq,v0)
vs1 = sol[0]
pprint("\nv0^2:")
pprint(latex(vs1*vs1))
pprint("\nv0:")
pprint(Num(vs1, 3))

print("\n--- c: -------------------------------")
half = S(1)/2
xM = half/g * vs1*vs1 * s2a
zM = half/g * vs1*vs1 * ss

pprint(Num(xM,3))
pprint(Num(zM,3))

print("\n--- d: -------------------------------")

twobeta = atan(xP/h)
pprint(Num(twobeta*180./pi,3))
beta = half*twobeta
pprint(Num(beta*180./pi,3))

cb = cos(beta)
tb = tan(beta)

z2 = -g/2/v0/v0/cb/cb *x*x + tb*x
rhs = z2.subs(x, 3*m)
eq = Eq(-h, rhs)

sol = solve(eq,v0)
vs2 = sol[0]
pprint("\nv:")
pprint(Num(vs2, 3))


print("\n--- e: -------------------------------")
pprint(Num(vs1,3))
pprint(Num(vs2,3))

# z in meters for alpha:
z_in_m_1 = z1.subs({v0:vs1})/m
z_in_m_2 = z2.subs({v0:vs2})/m

# z in m with argument xi = x/m:
xi = var("xi")
z1 = z_in_m_1.subs(x,xi*m).simplify()
z2 = z_in_m_2.subs(x,xi*m).simplify()

xia = linspace(float(0), float(3))

def fill_array(f,x,xia):
    if f.is_constant(x):
        res = full_like(xia,f)
    else:
        res = lambdify(x,f,"numpy")
        res = array(res(xia))
    return res
[z1a, z2a] = [fill_array(z,xi,xia) for z in [z1,z2]]

tmp = plt.axis('equal')
tmp = plt.grid()
tmp = plt.plot(xia,z1a, "b-", label=r"$\alpha$")
tmp = plt.plot(xia,z2a, "r--", label=r"$\beta$")
tmp = plt.xlabel(r"$x /  \mathrm{m}$")
tmp = plt.ylabel(r"$z \,/\, \mathrm{m}$")
tmp = plt.legend()
# plt.savefig("3.1.K_1.png", transp=True)
plt.show()

--- User input ----------------------- --- a: ------------------------------- \frac{1}{2} \frac{1}{4} \frac{\sqrt{3}}{2} \frac{3}{4} \frac{\sqrt{3}}{3} \frac{\sqrt{3}}{2} [None, None, None, None, None, None] - \frac{20 m x^{2}}{3 s^{2} v_{0}^{2}} + \frac{\sqrt{3} x}{3} --- b: ------------------------------- v0^2: \frac{60 m^{2}}{s^{2} \left(1 + \sqrt{3}\right)} v0: 4.69*m ------ s --- c: ------------------------------- 0.951*m 0.275*m --- d: ------------------------------- 71.6 35.8 v: 4.65*m ------ s --- e: ------------------------------- 4.69*m ------ s 4.65*m ------ s

--- User input ----------------------- --- a: ------------------------------- \frac{1}{2} \frac{1}{4} \frac{\sqrt{3}}{2} \frac{3}{4} \frac{\sqrt{3}}{3} \frac{\sqrt{3}}{2} [None, None, None, None, None, None] - \frac{20 m x^{2}}{3 s^{2} v_{0}^{2}} + \frac{\sqrt{3} x}{3} --- b: ------------------------------- v0^2: \frac{60 m^{2}}{s^{2} \left(1 + \sqrt{3}\right)} v0: 4.69*m ------ s --- c: ------------------------------- 0.951*m 0.275*m --- d: ------------------------------- 71.6 35.8 v: 4.65*m ------ s --- e: ------------------------------- 4.69*m ------ s 4.65*m ------ s