Non-Standard Trajectory Optimization (Part 2)

Classical Mechanics Level 5

A projectile is fired from ground level at an angle of θ \theta with respect to the horizontal. Let L L be the length of the arc of the projectile through the air. Let T T be the time the projectile stays in the air.

What value of θ \theta (in degrees) maximizes the product of L L and T T ?

Note: Assume an ambient downward gravitational field


The answer is 69.65.

Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
Feb 16, 2020

I took a pure brute-force approach to this. Sweep θ \theta in small increments. For each θ \theta , simulate the physics over time and keep track of the path length and flight time. Store the value of θ \theta which yields the greatest product. The answer is something close to 69.7 69.7 degrees. 

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import math

v0 = 10.0
g = 10.0

deg = math.pi/180.0

dtheta = deg/1000.0
dt = 10.0**(-5.0)

###################################

maxprod = 0.0

theta = 69.0*deg

while theta <= 70.0*deg:

    t = 0.0

    x = 0.0
    y = 0.0

    xd = v0*math.cos(theta)
    yd = v0*math.sin(theta)

    xdd = 0.0
    ydd = -g

    L = 0.0

    while y >= 0.0:

        dx = xd*dt
        dy = yd*dt

        dL = math.hypot(dx,dy)
        L = L + dL

        x = x + dx
        y = y + dy

        xd = xd + xdd*dt
        yd = yd + ydd*dt

        t = t + dt

    prod = t*L

    if prod > maxprod:

        maxprod = prod
        theta_store = theta/deg

    theta = theta + dtheta
    print (theta/deg),t

###################################

print ""
print ""
print theta_store

#69.668

2 Helpful 2 Interesting 2 Brilliant 0 Confused

@Legend of Physics Solution is up now

Steven Chase - 1 year, 3 months ago

@Steven Chase I thought that the peoples of USA are have good knowledge,and are educated and literate ,but after that incident in Congress ,I got to know ,that was wrong which I was thinking ,.

Talulah Riley - 5 months ago

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