Solving the final nonlinear equation in the problem

Question

Problem 8.2 in the textbook.

This problem has two parts:

The first part (the most important part) is solving the final nonlinear equation in the problem statement for θ. Once θ is known, you can this solve for the location of the football, i.e., solve for x and y using a series (or vector) of time values.

I used numpy.linspace() function to create a time vector and then looped through the time vector, solving for x and y at each time point.

Then, you can generate an x vs. y plot showing the trajectory of the ball. If the ball doesn't reach the target or if the ball goes through the ground (i.e., negative values for x) then you know that you choose a range of time values that is too short or too long.

If you've watched much (American) football, then you know that the ball is typically in the air for a few (more than 1 and less than 5 seconds) even for long throws.

Problem 8.2

The trajectory of any projectile object (under certain assumptions) can be determined using Newton's laws of motion, F = m.a. After a football has been released by a quarterback, the primary force acting on the football is gravity, thus, the acceleration, a, of the fotball can be described by:

O = m.a_x, -m.g = m.a_x

where a_x is acceleration in the x- or horizontal-direction, a_y is acceleration in y or vertical direction, m is the mass of the football, and g = 9.806 m/s². Since acceleration, a, equals the derivative of velocity with respect to time, dv/dt, the acceleration equations can be integrated to give equations for velocity:

v_x = v_o cos(θ)
v_y = v_o sin(θ) -g.t

where v_o is the initial velocity of the football and θ is the intial upward angle of the throw. Since velocity, v, equals the derivative of location, (x,y), with respect to time, the velocity equations can be integrated to give equations for location:

x  = x_o + v_ocos(θ).t

y = y_o + v_osin(θ).t - g.t²

where (x_o,y_o) is the initial location of the quarterback. Solving the x location equation for time gives:

t = (x - x_o)/ (v_ocos(θ))

and substituting this equation for time into the y location equation (and moving everything to one size of the equal sign gives:

O = y₀ + tan(θ).(x - x₀) - g.(x - x_o)²/(2.v_o²cos²(θ)) - Y

You have been hired by the quarterback of the local pi.iressional (American) football team to determine the angle, θ, for throwing the football so that it downfield for 50 meters and can be caught 2 meters above the ground. Use the following dat (3btain from the NFL Scouting Combine) in your analysis:

• Quarterback location (x_o,y_o) is (0.0,0.0)

• Quarterback arm strength gives an initial velocity of 25.0 m/s

• Target location (x,y) is (50.0,2.0) in In

• Broyden's method as implemented in Scipy is recommended

Your final report to the quarterback should include the initial angle, θ for the throw in radians and degrees, and you should have a plot of the ball location, (x,y) over time (e.g., Figure 8.5). Use this plot to estimate the time to target to the nearest second.