Projects for COS2000, Semester II, 2020/21 Project A2: Fr

Homework answers / question archive / Projects for COS2000, Semester II, 2020/21 Project A2: Free Fall, and SARS-Cov-2 spreading A Bungee & Superman Download and watch this video clip (Superman_saves_Lois_Lane, in the Projects folder on LumiNUS) from the 1978 movie version of Superman

Projects for COS2000, Semester II, 2020/21 Project A2: Free Fall, and SARS-Cov-2 spreading A Bungee & Superman Download and watch this video clip (Superman_saves_Lois_Lane, in the Projects folder on LumiNUS) from the 1978 movie version of Superman

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Projects for COS2000, Semester II, 2020/21

Project A2: Free Fall, and SARS-Cov-2 spreading

A Bungee & Superman

Download and watch this video clip (Superman_saves_Lois_Lane, in the Projects folder on LumiNUS) from the 1978 movie version of Superman. Lois Lane falls from the heliport atop the Daily Planet building. Superman catches her and saves the day.

1 Do some preliminary estimates based on the video:

(a) How long was Lois Lane in apparent free fall? Call this time T₁. (This doesn’t have to be very accurate; a lower and an upper value will be good enough.)

(b) How far would she have fallen in this time? Denote this fallen distance as D.

(d) How long did it take Superman to stop Lois Lane? Denote this time as T₂. Again, an estimate will do - see comment above.

For those of you who think you have forgotten everything about the physics of free fall, here is a one-line reminder: the distance (measuring from the point of fall) fallen under the constant gravitational force (acceleration g) at any time t (starting from t = O at the time of fall) is

X(t) = x₀ + u₀t + ½ gt²,

Where X₀ is the initial position (can be set up to be 0) and u₀ is the initial velocity/speed at t = 0. Can you figure out what the expression for the velocity v(t) is from here?

You can do more accurate estimates of times (especially of T₂): watch the video frame by frame. The video has a frame rate of about 30 frames per second, so each frame is of 1/30 second duration. Check out how you can do this in the free video player VLC: Https://www.vlchelp.com/frame-stepping/#:~: text=Just/20navigate/20in/20the7, 20timeline, one/,20frame/20at%20a/20time.

You can download the VLC media player here: https: //www.videolan.org/index. html.

X(t) = x₀ + u₀t + ½ gt²,

Where X₀ is the initial position (can be set up to be 0) and v₀ is the initial velocity/speed at t = 0. Can you figure out what the expression for the velocity v(t) is from here?

2 The bungee jump analogy:

(a) Lois Lane drops from a platform and free falls for a time of T₁ you estimated above. Take the estimated distance D she travels during this time to be the bungee cord length (L).

(b) The bungee cord is the Superman: it stretches and stops Lois Lane in the estimated time T₂.

(c) Run the bungee jumping code, adjusting the bungee cord elastic constant k to fit the rescue attempt. In other words, you want k such that the time between the moment the cord starts to stretch (x = L) to the instant Lois becomes stationary (v = 0). You might want to use plt.xlim(t1, t2) to zoom into the time interval (t1, t2) in the velocity plot.

Assume Lois Lane to have a mass of 60 kg.
Ignore air resistance.

(d) What is the maximum g force that Lois Lane experienced?

Comment on the likelihood of surviving a rescue attempt like this one. Notice that Lois Lane was fully conscious and her first words were "You got me? What’s got you?"

[In pulling out of a dive, for example, a trained pilot may be subjected to an acceleration as high as 9g. A typical person can handle 5g (meaning some people might pass out when riding a roller coaster, which in some cases exceeds this point) before losing consciousness.]

There is no “correct” answer here. The Key is whether you can make use of the observed (approximate) data and arrive at some conclusions and assessment of the scenario.

B SARS-CoV-2 modeling

A model for the SARS-CoV-2 is proposed in a recent paper:

https://journals.plos.org/plosone/article?id=10.1371/journal . pone .0236976

In this model, the population is divided into the following compartments:

S = susceptible individuals
A = asymptomatic infected individuals
I =symptomatic infected individuals
R=recovered individuals
D = individuals that died

The assumptions are:

(a) Upon exposure to the virus, some fraction (given by p) of people become asymptomatically infected, while the remaining (1 — p) fraction become symptomatically infected.

(b) The asymptomatic individuals can infect susceptible individuals, but with a different infection rate, determined by the infection rate c

(d) The asymptomatic individuals never develop symptoms and recover at the rate specified by k

(e) The symptomatic individuals recover at the rate specified by r

(f) Deaths come only from the symptomatic group at a rate specified by d

(g) The recovered individuals are no longer susceptible

(h) The populations are normalized, so that 1 £ S, A, I, R, D £ 1, and S + A + I + R + D =1

At any one time,

the susceptible individuals exposed to the virus is bSI + cSA - this is the rate of change in the susceptible population
the asymptomatic fraction is the p(bSI + cSA) (increase in the asymptomatic population), while the symptomatic fraction is (1— p)(bSI + cSA) (increase in the symptomatic population)
a fraction k of the asymptomatic, i.e. kA, recovers (decrease in the asymptomatic population, increase in the recovered population)
a fraction r of the symptomatic, i.e. rI, recovers (decrease in the symptomatic population, increase in the recovered population)
a fraction d of the symptomatic, i.e. dI, dies (decrease in the symptomatic population, increase in the dead population)

A graphical representation of the spread process is

Asymptomatic A ®^kA Recovered R

pS(bI + cA)

Susceptible S

(1 — p)S(bI + cA)

Symptomatic I ®^dI Dead D

By comparison to US data (California, Florida, New York, Texas), the approximate values of the parameters are:

Parameter	Value
b	0.52 – 1.12
c	0 – 2
d	0.056
K	0.143
R	0.143

(a) Set up the differential equations that govern the time evolution of the various compartmental population fractions.

(b) There is the suggestion that the parameter y should have a value greater than that of f.

What is your view? Why?

(c) Explore the impact of the different values of b = 0.5, 0.8, 1.1, c = 0.5, 1.0, 1.5, 2.0 and p = 0.3, 0.5, keeping the other parameters at the values from the US data above. (For a given b value, always use c values such that c > b.)

(d) Plot the population fractions and describe your observations.

(e) In particular, note in each case the peak of the infected fraction (asymptomatic + symptomatic) and the time of the peak, the (remaining) susceptible and dead fractions at the end of the simulation.

(f) Draw some conclusions from your observations.

March 9, 2021