Best way for a Fox Rabbit Chase Simulation in QBasic - algorithm

Question

I need some help with a homework.
Problem Definition
There is a rabbit 100 meters away from its hole and a fox 100 meters away from the rabbit in perpendicular direction to the hole.
(rabbit at (100,0), hole at (100,100), fox at (0,0)
The rabbit starts running straight to the hole with a given V1 speed and Fox chases the rabbit with a given V2 speed. Simulate the chase using QBasic and print if the rabbit gets caught or escapes.
I wrote some code, but it isn't working as it is supposed to. Even if the fox catches the rabbit, it prints that the rabbit escapes
My code so far:
CLS
SCREEN 12
WINDOW (-20, 120)-(120, -20)
LINE (0, 0)-(100, 0)
LINE (0, 100)-(0, 0)
CIRCLE (100, 100), 1
INPUT "Input speed of rabbit, v1=", v1
INPUT "input speed of fox, v2=", v2
dlt=0.01
x1 = 0: x2 = 0
y1 = 100: y2 = 0
drw: PSET (x1, y1), 1: PSET (x2, y2), 2
x1 = x1 + dlt * v1
x2 = x2 + dlt * v2 * (x1 - x2) / SQR((x1 - x2) ^ 2 + (y1 - y2) ^ 2)
y2 = y2 + dlt * v2 * (y1 - y2) / SQR((x1 - x2) ^ 2 + (y1 - y2) ^ 2)
IF SQR((x1 - x2) ^ 2 + (y1 - y2) ^ 2) < 0.01 GOTO caught
IF x1 > 100 GOTO escaped
GOTO drw
caught: PRINT "rabbit got caught": GOTO finish
escaped: PRINT "rabbit escaped"
finish: END
If you don't know about QBasic, you could also help me with an algorithm in any language. I just need a better algorithm to solve it.

Answer 1

First of all: although your assignment speaks of the rabbit moving along the Y-axis, you seem to have swapped X/Y coordinates throughout your code, so that actually the rabbit moves along the X axis in your implementation. This is a bit confusing, but it is not the cause of your problem.
The issue may be resulting from the margin of 0.01 you have in this test:
IF SQR((x1 - x2) ^ 2 + (y1 - y2) ^ 2) < 0.01 GOTO caught
If the speed of the fox is high enough, it may be that it "overshoots" the rabbit each time its new position is calculated, and end up at the other (alternating) side of the rabbit with a distance that is each time greater than 0.01. And so the above condition will never be true.
Instead test for the Y coordinate only (in your interpretation of X and Y), with:
IF y2 >= 100 GOTO caught
I also made an implementation in JavaScript (but with X and Y as given in the question), both with your "0.01" condition, and with the proposed fix. If you enter speeds V1=5 and V2=9, the fox can catch the rabbit, but you'll see the different outcome in both snippets:
Wrong:
async function animate() {
cls();
line(0, 0, 100, 0, "black");
line(0, 100, 0, 0, "black");
circle(100, 100, 3, "black");
let v1 = input("rabbit");
let v2 = input("fox");
let dlt = 0.1;
let x1 = 100, y1 = 0; // rabbit
let x2 = 0, y2 = 0; // fox
while (true) {
let y1to = y1 + dlt * v1;
let dist = Math.sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2);
let x2to = x2 + dlt * v2 * (x1 - x2) / dist;
let y2to = y2 + dlt * v2 * (y1 - y2) / dist;
line(x1, y1, x1, y1to, "green");
line(x2, y2, x2to, y2to, "red");
y1 = y1to;
x2 = x2to;
y2 = y2to;
// Problematic condition:
if (dist < 0.01) return output("rabbit got caught");
if (y1 >= 100) return output("rabbit escaped");
await delay(5);
}
}
let canvas = document.querySelector("canvas");
let ctx = canvas.getContext("2d");
// Some helper functions for JavaScript:
const input = (id) => +document.getElementById(id).value;
function line(x1, y1, x2, y2, color) {
ctx.beginPath();
ctx.moveTo(x1+0.5, y1+0.5);
ctx.lineTo(x2+0.5, y2+0.5);
ctx.strokeStyle = color;
ctx.stroke();
}
function circle(x, y, r, color) {
ctx.beginPath();
ctx.strokeStyle = color;
ctx.arc(x+0.5, y+0.5, r, 0, 2 * Math.PI);
ctx.stroke();
}
const output = (msg) => document.querySelector("#result").textContent = msg;
const delay = (ms) => new Promise(resolve => setTimeout(resolve, ms));
function cls() {
ctx.clearRect(0, 0, canvas.width, canvas.height);
output("");
}
// Bind a click handler for the button
document.querySelector("button").addEventListener("click", animate);
input { width: 4em }
.left { float: left; height: 180px; margin-right: 10px }
<div class="left">
<h3>Wrong implementation</h3>
Input speed of rabbit, v1= <input id="rabbit" type="number" value="5"><br>
Input speed of fox, v2= <input id="fox" type="number" value="9"><br>
<button>Go!</button><br>
</div>
<div>
<canvas width="105" height="105"></canvas>
<div id="result"></div>
</div>
Corrected:
async function animate() {
cls();
line(0, 0, 100, 0, "black");
line(0, 100, 0, 0, "black");
circle(100, 100, 3, "black");
let v1 = input("rabbit");
let v2 = input("fox");
let dlt = 0.1;
let x1 = 100, y1 = 0; // rabbit
let x2 = 0, y2 = 0; // fox
while (true) {
let y1to = y1 + dlt * v1;
let dist = Math.sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2);
let x2to = x2 + dlt * v2 * (x1 - x2) / dist;
let y2to = y2 + dlt * v2 * (y1 - y2) / dist;
line(x1, y1, x1, y1to, "green");
line(x2, y2, x2to, y2to, "red");
y1 = y1to;
x2 = x2to;
y2 = y2to;
// Corrected condition:
if (x2 >= 100) return output("rabbit got caught");
if (y1 >= 100) return output("rabbit escaped");
await delay(5);
}
}
let canvas = document.querySelector("canvas");
let ctx = canvas.getContext("2d");
// Some helper functions for JavaScript:
const input = (id) => +document.getElementById(id).value;
function line(x1, y1, x2, y2, color) {
ctx.beginPath();
ctx.moveTo(x1+0.5, y1+0.5);
ctx.lineTo(x2+0.5, y2+0.5);
ctx.strokeStyle = color;
ctx.stroke();
}
function circle(x, y, r, color) {
ctx.beginPath();
ctx.strokeStyle = color;
ctx.arc(x+0.5, y+0.5, r, 0, 2 * Math.PI);
ctx.stroke();
}
const output = (msg) => document.querySelector("#result").textContent = msg;
const delay = (ms) => new Promise(resolve => setTimeout(resolve, ms));
function cls() {
ctx.clearRect(0, 0, canvas.width, canvas.height);
output("");
}
// Bind a click handler for the button
document.querySelector("button").addEventListener("click", animate);
input { width: 4em }
.left { float: left; height: 180px; margin-right: 10px }
<div class="left">
<h3>Corrected implementation</h3>
Input speed of rabbit, v1= <input id="rabbit" type="number" value="5"><br>
Input speed of fox, v2= <input id="fox" type="number" value="9"><br>
<button>Go!</button><br>
</div>
<div>
<canvas width="105" height="105"></canvas>
<div id="result"></div>
</div>
A final remark on your code: the use of GOTO is frowned upon. You should instead use a DO WHILE loop for looping here, potentially with an EXIT statement if really needed.