Every falling leaf follows the perfect trajectory of gravity 🍁

Photo taken at Bayview Village Park, North York, Ontario, on November 4th, 2022.

Fall in Ontario is, without a doubt, my favourite time of year. The streets and parks transform into a golden blanket, becoming their most stunning before the pure white of snow removes all the colours. I love everything about fall: the taste of pumpkin in almost everything, from my mom’s famous traditional North Iranian dishes to the pumpkin lattes that fill every café. Fall offers the perfect palette of nature’s colours. It’s the time for beautiful styles—shawls paired with high boots—and, of course, my sister and grandpa's birthday (surprisingly on the same day!).

When I walk through the park, hearing the crunch of dried leaves under my feet and watching as gravity pulls them down, I find myself reflecting on how the trees are preparing for winter, getting ready for hibernation. In a way, as they prepare to sleep and shed their leaves, they give us this final burst of beauty before the cold sets. But beyond that, I also think of 9.81 m/s², the gravitational acceleration of our planet Earth. Will Earth always be our only home? Perhaps not for long, as we rush to reach far beyond, exploring and colonizing other worlds. Who knows what we’ll call Earth in the future—the "Old Planet," the "True Home," I’m not sure.

Where was I? Oh, right—let’s talk about the gravitational acceleration of Earth and how it affects the trajectory of autumn’s leaves. This is my review of Physics Grade 11, Chapter 2: 'Kinematics: Motion in Two Dimensions,' focusing on 2D Motion, for my students.

Assumption 1: Free Fall with No Initial Velocity

When we observe a leaf falling straight down with no initial velocity (meaning it simply drops due to gravity), it’s an example of free fall under Earth's gravitational acceleration.

Equation for Vertical Displacement (y):

Where:  y(t) is the vertical position of the leaf at time t. This equation shows that as time progresses, the leaf’s vertical displacement increases quadratically under the influence of gravity.

Assumption 2: Free Fall with an Initial Velocity Due to Wind (at an Angle)

If there's wind, the leaf might have an initial velocity at an angle θ. This creates a two-dimensional motion where the leaf moves both horizontally and vertically.

Vertical Motion:

Horizontal Motion:

Where: 

- Vo is the initial velocity due to the wind. 

- θ is the angle of the wind relative to the horizontal. 

- The term v0sinθ represents the vertical component of the initial velocity. 

- The term v0cosθ represents the horizontal component of the initial velocity.

In this scenario, the leaf moves in both directions, and its trajectory follows a parabolic path influenced by gravity and the initial wind angle.

Assumption 3: Introducing Air Resistance (Drag Force)

In real life, air resistance significantly affects the motion of falling leaves. Drag acts opposite to the direction of motion and depends on the leaf’s velocity, shape, and surface area. The drag force complicates the motion, making it non-linear.

Drag Force Equation:

Where: 

-Cd is the drag coefficient (depends on the shape of the leaf). 

- ρ is the air density. 

- A is the cross-sectional area of the leaf. 

- v is the speed of the leaf.

When drag is present, we need to solve differential equations for both the horizontal and vertical components:

  - The vertical velocity is influenced by gravity and drag. As time progresses, the leaf may reach a terminal velocity when the drag force equals the force of gravity.

Horizontal Motion with Drag:

  - The horizontal velocity decreases over time due to drag, meaning the leaf slows down horizontally.

Terminal Velocity:

  - This is the maximum velocity the leaf can reach when falling. The mass, drag coefficient, air density, and cross-sectional area all influence this value.

As we explore the movement of leaves, we can introduce additional factors to create a more accurate and realistic model. Let's focus on numerically solving the scenario involving drag force using a powerful tool like MATLAB.

We'll work with the following parameters to simulate the movement of a typical leaf:

-Mass: 0.001 kg

-Drag Coefficient (Cd​): 1.0 (reflecting the shape and texture of the leaf)

-Air Density (ρ): 1.225 kg/m^3 (at sea level conditions)

-Cross-sectional Area (A): 0.01 m^2

-Initial Velocity (v0​): 5 m/s (due to wind)

-Wind Angle (θ): 45 degrees

-Initial Height: 10 m (height from which the leaf falls)

Let's input these parameters into our MATLAB code and observe how it predicts the trajectory of the leaf, taking into account both gravity and air resistance.

% Parameters

g = 9.8; % Gravity (m/s^2)

m = 0.001; % Mass of the leaf (kg)

C_d = 1.0; % Drag coefficient (dimensionless, depends on shape)

rho = 1.225; % Air density (kg/m^3)

A = 0.01; % Cross-sectional area of the leaf (m^2)

v0 = 5; % Initial velocity (m/s, due to wind)

theta = pi / 4; % Initial angle of the wind (45 degrees)

initial_height = 10; % Initial height of the leaf (meters)

% Define the drag force equations (horizontal and vertical)

drag_force = @(v) (0.5 * C_d * rho * A * v^2);

% Define the system of ODEs for horizontal (vx) and vertical (vy) velocities

% z(1) = x, z(2) = vx, z(3) = y, z(4) = vy

dynamics = @(t, z) [...

z(2); % dx/dt = vx

-drag_force(z(2)) / m * sign(z(2)); % dvx/dt = drag force in x

z(4); % dy/dt = vy

-g - drag_force(z(4)) / m * sign(z(4))]; % dvy/dt = gravity + drag in y

% Initial conditions [x0, vx0, y0, vy0]

z0 = [0, v0 * cos(theta), initial_height, v0 * sin(theta)];

% Time span (0 to 10 seconds, or until object hits the ground)

tspan = [0 10];

% Solve the system using ode45

[t, z] = ode45(dynamics, tspan, z0);

% Extract x and y positions from solution

x = z(:, 1);

y = z(:, 3);

% Stop the simulation when the object hits the ground (y >= 0)

ground = find(y < 0, 1);

if ~isempty(ground)

x = x(1:ground);

y = y(1:ground);

end

% Set up the figure

figure;

axis([0 max(x)+1 0 initial_height + 1]);

xlabel('Horizontal Distance (m)');

ylabel('Vertical Height (m)');

title('Trajectory of a Falling Leaf with Wind and Air Resistance');

grid on;

hold on;

% Create VideoWriter object to save the animation as a WebM video

video = VideoWriter('FallingLeafSimulation.webm', 'MPEG-4');

video.FrameRate = 20; % Set the frame rate

open(video); % Open the video file for writing

% Plot leaf as a red diamond

leaf = plot(x(1), y(1), 'd', 'MarkerSize', 10, 'MarkerFaceColor', 'r', 'MarkerEdgeColor', 'r');

% Animation loop

for i = 1:length(t)

% Update the position of the leaf

set(leaf, 'XData', x(i), 'YData', y(i));

% Capture the current frame and write it to the video

frame = getframe(gcf);

writeVideo(video, frame);

% Pause for a short interval to create the animation effect

pause(0.05);

end

% Close the video file

close(video);

disp('Video saved as FallingLeafSimulation.webm');

Simulating the trajectory of a falling leaf with wind and air resistance using MATLAB.

To explore further, you can download MATLAB Academic and paste this code into it. Try changing the parameters—such as wind speed, angle, or the leaf’s shape—and observe how these adjustments affect the leaf's movement.

Happy Fall! 🍁🍂