Lesson 6-3: Cross-Disciplinary Modeling

Connect math to real problems in physics, economics, and biology.

Learning Objectives

No drag:

x(t)=v_0\cos\theta\, t

y(t)=v_0\sin\theta\, t-\tfrac{1}{2} g t^2

.Peak at

v_y=0

; eliminate

t

for range.

Demand

Q_d=a-bP

,Supply

Q_s=c+d(P-T)

;equilibrium solves

a-bP=c+d(P-T)

\dfrac{dN_1}{dt}=r_1 N_1\Bigl(1-\tfrac{N_1+\alpha N_2}{K_1}\Bigr)

\dfrac{dN_2}{dt}=r_2 N_2\Bigl(1-\tfrac{N_2+\beta N_1}{K_2}\Bigr)

Coexistence when

K_1/\alpha > K_2

and

K_2/\beta > K_1

Compute range and height for v0=20, θ=30°, g=9.8.

Solution:Range

R=\dfrac{v_0^{2}\sin(2\theta)}{g}\approx 35.3\,\text{m}

;max height

H=\dfrac{v_0^{2}\sin^{2}\theta}{2g}\approx 5.1\,\text{m}

Compute post-tax equilibrium with linear supply/demand.

Solution:Solve

a-bP=c+d(P-T)

→

P' = \dfrac{a-c+dT}{b+d}

Q' = a-bP'

;incidence depends on slopes

b,d

Problem 1

Compute range $R$ and max height $H$ for $v_0=20$ , $\theta=30^\circ$ , $g=9.8$ .

Solution: $R=\dfrac{v_0^{2}\sin(2\theta)}{g}\approx 35.3\,\text{m}$ , $H=\dfrac{v_0^{2}\sin^{2}\theta}{2g}\approx 5.1\,\text{m}$ .

Problem 2

Find post-tax equilibrium for $Q_d=100-2P$ , $Q_s=3(P-5)-50$ .

Solution: $P'=\dfrac{100+50+3\cdot 5}{2+3}=33$ , $Q'=100-2\cdot 33=34$ .

Problem 3

Competition: with $K_1=100$ , $\alpha=0.2$ , $K_2=80$ , $\beta=0.3$ , check coexistence conditions.

Solution: $K_1/\alpha=100/0.2=500>80$ , $K_2/\beta=80/0.3\approx 266.7>100$ → coexistence.

Problem 4

For fixed $v_0$ , show $\theta=45^\circ$ maximizes $R=\dfrac{v_0^{2}\sin(2\theta)}{g}$ (no drag).

Solution: maximize $\sin(2\theta)$ on $[0,\pi/2]$ → attained at $2\theta=\pi/2$ , i.e., $\theta=\pi/4=45^\circ$ .

Problem 5

Sensitivity: perturb $\alpha,\beta$ by ±10% and discuss how coexistence thresholds $K_1/\alpha, K_2/\beta$ change.

Solution: thresholds scale inversely with parameters; increasing $\alpha$ decreases $K_1/\alpha$ linearly, etc.

Return to Unit 6.