Modeling projectile motion subject to drag forces

PHYS 361Spring, 2011

projectile equations of motion

€

dvx

dt=

Fnet,x

m

2-D motion subject to gravity and drag forces

€

Fnet,x = Fdrag cosθ = −b2 v vx

Separate differential equations of motion

€

dvy

dt=

Fnet,y

m

x component

y component

€

Fnet,y = Fdrag sinθ − mg = −b2 v vy − mg

q

€

rv

€

vx€

vy

If we want to solve for position, also... we need two more equations

€

dx

dt= vx

€

dy

dt= vy

Euler method

€

dx

dt≈

Δx

Δt=

x i+1 − x i

Δt

x-position

y-position

x-velocity

y-velocity€

dvx

dt=

−b2 v vx

m

€

dvy

dt=

−b2 v vy − mg

m

€

dx

dt= v

€

dy

dt= w €

x i+1 = x i + v iΔt

€

y i+1 = y i + wiΔt

€

v i+1 = v i − Δtb2

mv i v i

2 + wi2

€

wi+1 = wi + Δt −g −b2

mwi v i

2 + wi2 ⎛

⎝ ⎜

⎞

⎠ ⎟

first-order approximation of derivative

programming efficiencies

€

x i+1 = x i + v iΔt

€

y i+1 = y i + wiΔt

€

v i+1 = v i − Δtb2

mv i v i

2 + wi2

€

wi+1 = wi + Δt −g −b2

mwi v i

2 + wi2 ⎛

⎝ ⎜

⎞

⎠ ⎟

1. pre-define constants that get used in for/while loops

2. rename lengthy expressions that have a particular significance

€

f =b2

mv i v i

2 + wi2