Question

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.

Answer 1

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