Elastic Collision in a Pool Game

Posted on February 7, 2023 by Administrator Posted in Computer Science, Computing Concepts, JavaScript, Solved Challenges

A range of video games use elastic collision formulas to predict the change of velocity of two objects when a collision occurs.

Elastic collision occurs when two objects are colliding and the total kinetic energy of the two objects remains the same. In reality, most collisions between two objects would result in some loss of energy (inelastic collision). However in some contexts, such as when two rigid billiard balls are colliding, this loss of energy is negligeable. In a video game of pool we would therefore implement an algorithm based on an elastic collision.

Let’s investigate how we can implement an elastic collision in a computer animation or a video game.

Velocity Vector

In a frame based game, most moving sprites would have a set of coordinates (x,y) to inidicate their position on the screen as well as a vaolcity vector (vx,vy) to indicate used to increment the (x,y) coordinates between two frames of the game and hence implement the movement of the sprite.

1 Dimensional Elastic Collision

Let’s consider two perfectly elastic balls of masses m1 and m2 moving along the same straight line with velocities u1 and u2.

Our aim is to calculate the velocity v1 and v2 of these two balls after the collision.

Conisidering that both momentum and kinetic energy are conserved quantities in an elastic collision, we can deduct the following two formulas:

Conservation of momentum

Conservation of kinetic energy

We can use both these formulas to calculate the velocity vectors v1 and v2 of the two colliding objects after the collision:

Note that these equations can be simplified when both colliding objects have the same mass: m1=m2. In this case we can use the following simplified formulas:

2 Dimensional Elastic Collision

In a 2-dimension environment, the velocity vectors may not be aligned on the same straight line.

We can however decompose the velocity vectors to identify the component of the velocity that is going along the straight line joining the centre of both moving balls (Purple vectors on the above diagram) and the component that is perpendicular to this straight line (Grey vectors on the above diagram). The first component will be affected by the elastic collision using the 1-Dimensional model/formulas whereas the second component will not be affected by the collision.

In order to calculate the 2 components of our velocity vectors (purple and grey vectors) we will first need to perform a rotation by θ, the angle that can be calculated using the (x,y) cartesian coordinates of the centre of both moving objects as follows:

We will then rotate our velocity vectors by θ as shown on the diagram below. Note however, that we would apply the same rotation for both velocity vectors (red and blue balls).

The formulas to perform a 2D rotation are as follows: (You can find out more about these formulas on this page)
where (V_x,V_y) represents the velocity vector (v_x,v_y) after the rotation.

We can now apply the 1-dimensional elastic collision formulas to the V_x (purple component) of the velocity for each moving object, whereas the V_y components will not be affected by the elastic collision.

And then we will need to rotate our new velocity vectors by -θ to cancel out the previous rotation.

2-Dimensional Elastic Collision Demonstration

To see how the above formulas can be implemented, we have created a demo using JavaScript. You can investigate this code further to identify how the steps described above have been implemented.

In our demo, all the moving objects have a different mass, pro-rata of their size (radius).

Note that this code could be simplified further using objects of the same mass as this would be the case in a game of pool!

Finally, you will notice that this code also applies some formulas to the velocity vectors of each ball to let the balls bounce against the edge of the canvas. These formulas are explained on this page.
<br /> See the Pen <a href="https://codepen.io/101Computing/pen/jOpdWgr"><br /> Elastic Collision in 2D</a> by 101 Computing (<a href="https://codepen.io/101Computing">@101Computing</a>)<br /> on <a href="https://codepen.io">CodePen</a>.<br />

Your Task

Your task consists of tweaking the above code (Click on “Edit On Codepen” button in the top right corner of the above codepen frame) in order to create a pool table with:

7 yellow balls
7 red balls
1 black ball
1 white ball

Note that, in a game of pool, all balls have the same mass and size. You can hence simplify the code to use the simplified 1-dimensional elastic collision formulas.

About this task: The aim of this challenge is not to create a full game of pool but just to tweak the above code to make sure that the canvas contains 16 balls of the right colour and of the same weight/size. In a full game of pool, other features would need to be considered such as, implementing the correct size of the pool table, adding pockets and detecting when balls fall into these pockets, adding friction to slow down the rolling balls and adding a mechanism for the player to aim/shoot. All of these features are not part of this task.

Solution...

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The Monty Hall Problem

Posted on February 2, 2023 by Administrator Posted in Computer Science, Python - Advanced, Python Challenges, Solved Challenges

The Monty Hall problem is a counter-intuitive brain teaser based on probabilities.

This puzzle is named after Monty Hall, the TV presenter of “Let’s Make a Deal”, an American TV show (known as “Deal or No Deal” in the UK) where contenders swap boxes of different values in order to win a prize.

In a similar approach, the aim of the Monty Hall puzzle is to try to win a car by knocking on the right door. The player is presented with three closed doors. Behind one of these doors is a luxurious car. Behind each of the other two doors is a goat!

Here is how the game proceeds:

You first pick a door. It could be that you picked the door with the car but you do not know that yet!
The game show host examines the other two doors and opens one with a goat. (Note that when both doors are hiding a goat, the presenter picks one of these doors randomly.)
You then need to make a decision: You can either stick with your initial guess and open the door to see if you win the car. Alternatively, you can decide to switch your door with the other closed door!

You can give it a go using our interactive game below:

What are the Odds?

When being asked whether to stick to your first guess or change your mind, there are only two closed doors left: One door hiding a goat and one door hiding the car. You could therefore assume that the odds of winning the car are 50-50! This is however a misconception! And if you play the game several times in a row, you will soon realise that you are twice more likely to win the car if you decide to switch your initial guess with the other door!

Not convinced? Try it… And if you want to fully understand the probabilities of this counter-intuitive puzzle you can check this page.

Frequency Analysis

In order to confirm these odds, we would like you to create a Python program that will simulate 100 games in a row, randomly positioning the two goats and the car behind the three doors every time. The program should then automatically and randomly select a door, reveal one door/goat and decide whether to switch doors or not. It should then determine whether the selected door leads to the car or the goat and keep a tally count of wins and losses.

At the end of the 100 attempts, the program should output the statistics as follows:

Python Code

We have only just started the code and thought about the key steps needed to complete this challenge. Your task is to complete the code below to produce a similar output (See statistics displayed above).

Solution...

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Denary to Binary Conversion Algorithm

Posted on January 31, 2023 by Administrator Posted in Computer Science, Python - Intermediate, Python Challenges, Solved Challenges

In this challenge, we are going to write al algorithm to convert a decimal (aka Denary) number between 0 and 255 into binary using 1 Byte (=8 bits).

Before attempting this challenge you may practise your denary to binary conversion:

Python Challenge

We are now going to use some Python code to implement al algorithm to convert a denary number into binary. To do so, you will need to complete the following 4 tasks.

Python Challenge - Task #1Task #2: Input ValidationTask #3: Writing a FunctionTask #4: Binary to Denary

Python Challenge: Task #1

We have started an algorithm which we would like you to complete to perform a full denary to binary conversion.

Your first challenge is to investigate the code below to see what it does and how it works. You will then need to complete this code to make sure that it outputs a fully Byte of data for any given number between 0 and 255.

The following trace table may help you investigate this code step by step:

	Line Number	denary	binary	OUTPUT
	3

Python Challenge: Task #2

Add some code to your code from task 1 to validate the input so that your code only accepts a number between 0 and 255. You can find out more about different validation techniques below and select the validation technique that is most relevant to this task:

Validation Techniques:

Input Validation: Presence Check

name = input("Enter your name:").strip() 

if name=="":
    print("Empty name!")
else:
    print("Thank you!")

Input Validation: Type Check – Integer?

number = input("Type a number:")

if number.isdigit():
    print("This is a number")
else:
    print("This is not a whole number")

Input Validation: Range Check

number = int(input("Type a number between 1 and 5:"))

if number>=1 and number<=5:
    print("Valid number")
else:
    print("Invalid number")

Input Validation: Lookup Check

drive = input("Can you drive?").lower()

if drive in ["yes","no"]:
    print("Valid answer")
else:
    print("Invalid answer")

Input Validation: Character Check

email = input("Type your e-mail address:")

if "@" in email:
    print("Valid e-mail address")
else:
    print("Invalid e-mail address")

postcode = input("Type your postocode:").upper()

if postcode[0] in "ABCDEFGHIJKLMNOPQRSTUVWXYZ" and postcode[1] in "ABCDEFGHIJKLMNOPQRSTUVWXYZ" and postcode[2] in "123456789":
    print("Valid postcode")
else:
    print("Invalid postcode")

Input Validation: Length Check

password = input("Type a password:")

if len(password)>=8:
    print("Valid password")
else:
    print("Invalid password")

Try Again! Using a While Loop:

name = input("Enter your name:")

while name=="":
    print("You must enter your name! Please try again!")
    name = input("Enter your name:")

print("Welcome " +  name)

Task #3: Writing a Function

Turn your code into a function called convertToBinary(). Your function will take a denary number as a parameter and return a binary string. You will then need to add some code to test your function.

Task #4: Writing a Function

Create a new function called convertToDenary(). This new function will take a binary string as a parameter and return the matching denary value.

Solution...

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Software Crosswords

Posted on January 17, 2023 by Administrator Posted in Computer Science, Computing Concepts, Computing Terminology Quiz, Crosswords, GCSE Concepts, GCSE Quiz, Solved Challenges

A computer system consists of both hardware (the physical components of the computer) and software: the programs that run on this hardware.

There are three main categories of software:

Operating Systems: software that controls the computer hardware: The operating system can load programs into memory (e.g. RAM), instruct the CPU on which programs to run, interact with input, output and storage devices.
Utilities and Drivers: A range of specific software that can be used to help with the maintenance and security of the computer system
Application Software: A range of software used by the end-user to perform specific tasks.

Complete the following crossword to demonstrate your understanding of the three main types of computer software.
Computer Software CrosswordOpen Crossword Puzzle in a New Window
Computer Software CrosswordOpen Crossword Puzzle in a New Window

Solution...

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Short Path Algorithm Practice

Posted on January 10, 2023 by Administrator Posted in A Level Concepts, A Level Quiz, Algorithms, Computer Science, Computing Concepts

Before completing this task, you will need to familiarise yourself with the following 2 algorithms used to find the shortest path between two nodes of a weighted graph:

Dijkstra’s Short Path Algorithm

For each of the weighted graph below, complete the table below to show the steps needed to find the shortest path between node A and node Z using the Dijkstra’s Short Path Algorithm. Note that on these graphs, the weight of each edge represents the distance in miles between two nodes.

Graph #1Graph #2Graph #3Graph #4