- import numpy as np
- # Define the velocity function v(t)
- def v(t):
- return -0.00002*t**5 + 0.00267*t**4 - 0.11383*t**3 + 1.95292*t**2 - 10.39833*t
- # Define the derivative of the velocity function v'(t)
- def v_prime(t):
- return -0.0001*t**4 + 0.01068*t**3 - 0.34149*t**2 + 3.90584*t - 10.39833
- # Define the integrand for the length of the roller coaster track L
- def integrand_length(t):
- return np.sqrt(1 + v_prime(t)**2)
- # Define the integrand for the total distance traveled D
- def integrand_distance(t):
- return np.abs(v(t))
- # Define the trapezoidal rule integration function
- def trapezoidal_rule_integration(func, a, b, n=1000):
- h = (b - a) / n
- x = np.linspace(a, b, n+1)
- y = func(x)
- return h * (0.5*y[0] + 0.5*y[-1] + np.sum(y[1:-1]))
- # Define the interval [a, b]
- a = 0
- b = 50
- # Approximate the integral for the length of the roller coaster track L
- length_approx = trapezoidal_rule_integration(integrand_length, a, b)
- # Approximate the integral for the total distance traveled D
- distance_approx = trapezoidal_rule_integration(integrand_distance, a, b)
- print("Approximate length of roller coaster track:", length_approx)
- print("Approximate total distance traveled by rider:", distance_approx)
[text] maths
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