Modeling and Simulation in Python

Chapter 23

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International

In [1]:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline

# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

# import functions from the module
from modsim import *

Code from the previous chapter

In [2]:
m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram
degree =

In [3]:
t_end = 20 * s
dt = t_end / 100

params = Params(x = 0 * m, 
                y = 1 * m,
                g = 9.8 * m/s**2,
                mass = 145e-3 * kg,
                diameter = 73e-3 * m,
                rho = 1.2 * kg/m**3,
                C_d = 0.3,
                angle = 45 * degree,
                velocity = 40 * m / s,

In [4]:
def make_system(params):
    """Make a system object.
    params: Params object with angle, velocity, x, y,
               diameter, duration, g, mass, rho, and C_d
    returns: System object
    angle, velocity = params.angle, params.velocity
    # convert angle to degrees
    theta = np.deg2rad(angle)
    # compute x and y components of velocity
    vx, vy = pol2cart(theta, velocity)
    # make the initial state
    R = Vector(params.x, params.y)
    V = Vector(vx, vy)
    init = State(R=R, V=V)
    # compute area from diameter
    diameter = params.diameter
    area = np.pi * (diameter/2)**2
    return System(params, init=init, area=area)

In [5]:
def drag_force(V, system):
    """Computes drag force in the opposite direction of `V`.
    V: velocity Vector
    system: System object with rho, C_d, area
    returns: Vector drag force
    rho, C_d, area = system.rho, system.C_d, system.area
    mag = rho * V.mag**2 * C_d * area / 2
    direction = -V.hat()
    f_drag = direction * mag
    return f_drag

In [6]:
def slope_func(state, t, system):
    """Computes derivatives of the state variables.
    state: State (x, y, x velocity, y velocity)
    t: time
    system: System object with g, rho, C_d, area, mass
    returns: sequence (vx, vy, ax, ay)
    R, V = state
    mass, g = system.mass, system.g
    a_drag = drag_force(V, system) / mass
    a_grav = Vector(0, -g)
    A = a_grav + a_drag
    return V, A

In [7]:
def event_func(state, t, system):
    """Stop when the y coordinate is 0.
    state: State object
    t: time
    system: System object
    returns: y coordinate
    R, V = state
    return R.y

Optimal launch angle

To find the launch angle that maximizes distance from home plate, we need a function that takes launch angle and returns range.

In [8]:
def range_func(angle, params):  
    """Computes range for a given launch angle.
    angle: launch angle in degrees
    params: Params object
    returns: distance in meters
    params = Params(params, angle=angle)
    system = make_system(params)
    results, details = run_ode_solver(system, slope_func, events=event_func)
    x_dist = get_last_value(results.R).x
    print(angle, x_dist)
    return x_dist

Let's test range_func.

In [9]:
range_func(45, params)

And sweep through a range of angles.

In [10]:
angles = linspace(20, 80, 21)
sweep = SweepSeries()

for angle in angles:
    x_dist = range_func(angle, params)
    sweep[angle] = x_dist

Plotting the Sweep object, it looks like the peak is between 40 and 45 degrees.

In [11]:
plot(sweep, color='C2')
decorate(xlabel='Launch angle (degree)',
         ylabel='Range (m)',
         title='Range as a function of launch angle',


We can use maximize to search for the peak efficiently.

In [12]:
bounds = [0, 90] * degree
res = maximize(range_func, bounds, params)

res is an ModSimSeries object with detailed results:

In [13]:

x is the optimal angle and fun the optional range.

In [14]:
optimal_angle = res.x

In [15]:
max_x_dist =

Under the hood

Read the source code for maximize and minimize_scalar, below.

Add a print statement to range_func that prints angle. Then run maximize again so you can see how many times it calls range_func and what the arguments are.

In [16]:

In [17]:

The Manny Ramirez problem

Finally, let's solve the Manny Ramirez problem:

What is the minimum effort required to hit a home run in Fenway Park?

Fenway Park is a baseball stadium in Boston, Massachusetts. One of its most famous features is the "Green Monster", which is a wall in left field that is unusually close to home plate, only 310 feet along the left field line. To compensate for the short distance, the wall is unusually high, at 37 feet.

Although the problem asks for a minimum, it is not an optimization problem. Rather, we want to solve for the initial velocity that just barely gets the ball to the top of the wall, given that it is launched at the optimal angle.

And we have to be careful about what we mean by "optimal". For this problem, we don't want the longest range, we want the maximum height at the point where it reaches the wall.

If you are ready to solve the problem on your own, go ahead. Otherwise I will walk you through the process with an outline and some starter code.

As a first step, write a function called height_func that takes a launch angle and a params as parameters, simulates the flights of a baseball, and returns the height of the baseball when it reaches a point 94.5 meters (310 feet) from home plate.

In [18]:
# Solution goes here

Always test the slope function with the initial conditions.

In [19]:
# Solution goes here

In [20]:
# Solution goes here

Test your function with a launch angle of 45 degrees:

In [21]:
# Solution goes here

Now use maximize to find the optimal angle. Is it higher or lower than the angle that maximizes range?

In [22]:
# Solution goes here

In [23]:
# Solution goes here

In [24]:
# Solution goes here

With initial velocity 40 m/s and an optimal launch angle, the ball clears the Green Monster with a little room to spare.

Which means we can get over the wall with a lower initial velocity.

Finding the minimum velocity

Even though we are finding the "minimum" velocity, we are not really solving a minimization problem. Rather, we want to find the velocity that makes the height at the wall exactly 11 m, given given that it's launched at the optimal angle. And that's a job for root_bisect.

Write an error function that takes a velocity and a Params object as parameters. It should use maximize to find the highest possible height of the ball at the wall, for the given velocity. Then it should return the difference between that optimal height and 11 meters.

In [25]:
# Solution goes here

Test your error function before you call root_bisect.

In [26]:
# Solution goes here

Then use root_bisect to find the answer to the problem, the minimum velocity that gets the ball out of the park.

In [27]:
# Solution goes here

In [28]:
# Solution goes here

And just to check, run error_func with the value you found.

In [29]:
# Solution goes here

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