Prof. Szuberla's Homework #3 [Solution] § PHYS 220, Spring 2009

PHYS 220 — HW #3 [Solution]

This assignment is due at the beginning of lecture on Monday, 16 Feb.

In lecture we've shown how an n^th-order ode can be recast as a system of n 1^st-order ode. In these 4 problems, you'll be translating the ode given into a single line of MATLAB code. I'll give an example first:

The 2^nd-order ode d²x/dt² = -ω²x for a simple harmonic oscillator
can be recast as dx/dt = v and dv/dt = -ω²x
a MATLAB vector can then be constructed to represent the variables, figuratively, as y = [ x ; v ]
and the ode can be written, literally, in MATLAB as dydx = [ y(2) ; -omega^2 * y(1) ];

For each of the ode below, write both the figurative construction of y = [ ... ], and the literal line of MATLAB code corresponding to dydx = [ ... ]. (In essence, you're writing the function that the function handle deriv will point to.)The solutions below are not unique, since there are many ways to represent both your figurative variable from the physics/mathematics, and another many ways to implement the code in MATLAB. What you see below are reasonable solutions.

In simulating a bicyclist capable of constant power output P, we find dv/dt = F/m, with F_b = P/v & F_drag = -av - bv². (This is a 1^st-order ode problem, so do not assume F = md²x/dt².)
Figurative: y = v
MATLAB: dydt = 1/m*( P/v - a*v - b*v^2);

Note that since this is a scalar problem, the brackets are unnecessary.

A damped, non-linear simple pendulum has an equation of motion d²θ/dt² = -Ω² sin θ - q dθ/dt.
Figurative:
y = [
th
w
]

MATLAB:
dydt = [
y(2)
-Omega^2*sin(th) - q*y(2)
];

For a mass m in free fall without air resistance near the surface of a planet, we find F_x = 0 & F_y = -mg. (Here you do need to recognize that this is a system of 2^nd-order ode.)
Figurative:
y = [
y
v_y
]

MATLAB:
dydt = [
y(2)
-g
];

In projectile motion, one might find equations of motion for the projectile as F_x = -mbvv_x & F_y = -mg -mbvv_y. (Here v is the magnitude of the velocity vector and v_i are its components.)
Figurative:
y = [
x
y
v_x
v_y
]

MATLAB:
vx = y(3);
vy = y(4);
magV = sqrt(vx^2 + vy^2);
dydt = [
vx
vy
-(b*magV*vx)/m
-g - (b*magV*vy)/m
];