Use Python to solve each problem.
2. A particle moves according to a law of motion s = t 3 − 12t 2 + 24t, t ≥ 0.
a) Find the velocity at time t.
b) What is the velocity after 1 second?
c) When is the particle at rest?
d) Sketch the position function on t ∈ [0, 6] to determine when the particle is moving in the positive direction on that interval.
e) Find the total distance traveled in the first 6 seconds (exact and approximate).
f) Find the acceleration after 1 second and at the times found in part c).
g) Graph the position, velocity, and acceleration functions on the same set of axes for t ∈ [0, 8].
Please use Python commands.
from sympy import *
from numpy import arange
import numpy as np
import matplotlib.pyplot as plt
t = Symbol('t')
#t 3 − 12t 2 + 24t
s = t**3 -(12*t**2) + (24*t)
sprime = s.diff(t)
print(sprime)
f = lambdify(t, sprime, 'numpy')
a = np.array([1])
print(f(a))
#solve for sprime =0 using solve function part c
roots = solve(sprime, t)
print(roots)
# d
#position at t = 6
f = lambdify(t, s, 'numpy')
a = np.array([6])
print(f(a))
#acceleration is derivative of velocity
sprimedouble = sprime.diff(t)
#display acceleration
print(sprimedouble)
#find value at 1
f = lambdify(t,sprimedouble , 'numpy')
a = np.array([1])
print(f(a))
a = np.array(roots)
print(f(a))
#plot position part d
t = np.arange(0.0, 6.0, 0.01)
s = (t**3 -(12*t**2) + (24*t))
plt.plot(t, s)
plt.xlabel('time')
plt.ylabel('Position')
plt.grid(True)
plt.savefig("positio.png")
plt.show()
#part g all 3 together
t = np.arange(0.0, 8.0, 0.01)
s = (t**3 -(12*t**2) + (24*t))
sprime = (3*t**2 -(24*t) + (24))
plt.plot(t, s)
sprimedouble = (6*t) - 24
plt.xlabel('time')
plt.ylabel('Position, velocity, acceleration')
plt.grid(True)
plt.savefig("positio.png")
plt.show()
#if you save as same fig it will overlap them on the same axis
plt.plot(t, sprime)
sprimedouble = (6*t) - 24
plt.savefig("positio.png")
plt.show()
plt.plot(t, sprimedouble)
plt.savefig("positio.png")
plt.show()
plot
last plot for all values
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