Shot Put Simulation with Pygame

The key physical quantities in projectile motion are:

• Initial velocity \( v_0 \): The speed at which the object is thrown.

• Launch angle \( \theta \): The angle at which the object is thrown relative to the horizontal axis.

• Acceleration due to gravity \( g \approx 9.81 \, \text{m/s}^2 \).

• Time \( t \): The time elapsed since the throw.

Here are the formulas for all four important parameters of projectile motion: flight time, maximum height, range, and final velocity.

1. Flight Time (\( t_{\text{total}} \))

The total flight time is the duration the ball remains in the air until it reaches the ground again.

\[ t_{\text{total}} = \frac{2 \cdot v_0 \cdot \sin(\theta)}{g} \]

• \( v_0 \) = initial velocity (m/s)
• \( \theta \) = launch angle (in degrees)
• \( g \) = gravitational acceleration (approx. 9.81 m/s²)

2. Maximum Height (\( y_{\text{max}} \))

The maximum height is the highest position the ball reaches before falling back down.

\[ y_{\text{max}} = \frac{(v_0 \cdot \sin(\theta))^2}{2 \cdot g} \]

Here, only the vertical component of the initial velocity \( v_0 \cdot \sin(\theta) \) is used because it is solely responsible for the change in height.

3. Range (\( x_{\text{max}} \))

The range is the horizontal distance the ball covers until it hits the ground.

\[ x_{\text{max}} = \frac{v_0^2 \cdot \sin(2\theta)}{g} \]

• The term \( \sin(2\theta) \) shows that the maximum throw occurs at an angle of \( \theta = 45^\circ \).

4. Final Velocity

The final velocity can be split into two components: the horizontal velocity (which remains constant) and the vertical velocity (which changes due to gravity).

Horizontal Final Velocity:

\[ v_{x} = v_0 \cdot \cos(\theta) \]

Vertical Final Velocity:

Vertical velocity changes because of gravitational acceleration. At the endpoint (impact), the vertical final velocity is: \[ v_{y} = v_0 \cdot \sin(\theta) - g \cdot t_{\text{total}} \]

Total Final Velocity (\( v_{\text{end}} \)):

The total final velocity can be computed with the Pythagorean theorem since it is the combination of horizontal and vertical velocities:

\[ v_{\text{end}} = \sqrt{v_{x}^2 + v_{y}^2} \]

This gives the velocity of the ball immediately before it hits the ground.

Pygame Simulation

import pygame
import math

# Initialisiere Pygame
pygame.init()

# Fenstergröße
width, height = 800, 600
screen = pygame.display.set_mode((width, height))
pygame.display.set_caption('Kugelwurf Simulation')

# Farben
WHITE = (255, 255, 255)
RED = (255, 0, 0)

# Physikalische Konstanten
g = 9.81  # Erdbeschleunigung in m/s²

# Kugelparameter
x0, y0 = 50, height - 50  # Startposition (linke untere Ecke)
v0 = 100  # Startgeschwindigkeit in Pixel/s
angle = 45  # Wurfwinkel in Grad
v0x = v0 * math.cos(math.radians(angle))  # Horizontale Geschwindigkeit
v0y = v0 * math.sin(math.radians(angle))  # Vertikale Geschwindigkeit
t = 0  # Zeit in Sekunden
dt = 1 / 25  # Zeitschritt in Sekunden (25 Frames pro Sekunde)

# Clock-Objekt, um die FPS zu kontrollieren
clock = pygame.time.Clock()

# Simulations-Loop
running = True
while running:
    screen.fill(WHITE)

    # Beende das Programm, wenn das Fenster geschlossen wird
    for event in pygame.event.get():
        if event.type == pygame.QUIT:
            running = False

    # Kugelposition basierend auf den Gleichungen des Kugelwurfs
    x = x0 + v0x * t
    y = y0 - (v0y * t - 0.5 * g * t ** 2)

    # Zeichne die Kugel
    pygame.draw.circle(screen, RED, (int(x), int(y)), 10)

    # Aktualisiere das Display
    pygame.display.flip()

    # Aktualisiere die Zeit
    t += dt

    # Kontrolliere die Bildwiederholrate (25 FPS)
    clock.tick(25)

    # Beende die Simulation, wenn die Kugel den Boden erreicht
    if y > height - 50:
        running = False

# Beende Pygame
pygame.quit()