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MATH36031 Diminishing speeds

创建日期：2024-08-15 15:26:46

所属分类：数学mathematics

标签： MATH36031

Diminishing speeds

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MATH36031 Project

In this project, you will investigate the problem of a fox chasing a rabbit by solving differential

equations modelling their positions at different times. The initial configuration is shown in

Figure 1, where the fox starts chasing the rabbit while the rabbit tries to escape from its

predator and moves towards its burrow instantaneously. The fox is initially located1 at

(?250,?550), pursues the rabbit with the initial speed sf0 = 16m/s in one of the following

two possible ways:

Warehouse

F(-250,-550)

S(-200,-400)

R(0,0)

B(-600,600)

N(-200,0)

Figure 1: Coordinates (in metres) of the fox (F), the rabbit (R) and its burrow (B) and the

two corners (S,N) of the warehouse.

if the rabbit is in sight, the fox’s attack path points directly towards the rabbit (the

direction of the velocity vector of the fox is exact from the fox to the rabbit);

if the view of the rabbit is blocked by the corner S of an impenetrable warehouse

(assuming that the warehouse is extended indefinitely to the west), then the fox runs

directly towards this corner. If the rabbit is still not in sight when the corner is

reached, then the fox moves parallel to the N-S perimeter of the warehouse until it sees

the rabbit.

1The subscripts f and r denote the fox and rabbit respectively. The units of the coordinates are metres.

The rabbit, initially located at the origin (0, 0), runs towards its burrow at (?600, 600)

in a straight line with initial speed sr0 = 13m/s.

Question 1: Constant speeds.. Assuming that both the fox and the rabbit run with

constant speeds sf0 = 16m/s and sr0 = 13m/s respectively, determine whether the rabbit

can be captured before it reaches its burrow. The rabbit is considered to be captured by the

fox, if the distance between them is smaller than or equal to 0.1 meter.

Question 2: Diminishing speeds. Let us consider a more realistic scenario, when

the hungry fox meets the tired rabbit. Because neither the fox nor the rabbit are in their

best conditions, their chasing/escaping speeds diminish in time, according to the amount of

distance (starting from the time they find each other and start running) they have travelled

so far. More precisely, their speeds at time t are given by

sf (t) = sf0e

μfdf (t), sr(t) = sr0e?μrdr(t),

where sf0 = 16m/s and sr0 = 13m/s are the same initial speeds as above, μf = 0.0002m

and μr = 0.0008m

1 are the rates of the diminishing speeds, df (t) and dr(t) are the distance

they have travelled up to time t(> 0). Determine whether the rabbit can be captured before

it reaches its burrow. (You may assume that this diminishing speed starts from t = 0).

Outputs required You are required to submit a report (maximum 8 pages including

any appendices) in pdf form via the Turnitin submission box on Blackboard. Additionally

you need to submit your m-files used for the MATLAB codes via the Blackboard submission

box. In your report give (i) the time T and the location of the fox when either the rabbit is

captured or the rabbit escapes to the burrow; (ii) the distance travelled by the fox in time

T . Additionally provide a plot showing the paths taken by both animals.

Additional information and guidelines

1. All coding must be done in MATLAB.

2. Treat both the fox and the rabbit as points, without worrying about their finite sizes

(as in most models).

3. The questions can be answered with different approaches, but the preferred way is to

use the built-in ODE solver ode45 discussed during the lectures.

4. Avoid using hard-coded numbers. Any number in your code should either be given as

initial condition, or be derived from these conditions.

5. Keep your page length not exceeding eight A4 pages, with a font size no smaller

than 11, and page margins no smaller than 2cm. There is no need for a title page for

a relative short report like this. If more than 8 pages are submitted only the

first 8 pages will be marked and the rest of the submission will be ignored.

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