Projectile Launcher#

Welcome to the Projectile Launcher!#

Have you ever wondered what happens when you throw a ball into the air? Or how a cannonball flies through the sky? The Projectile Launcher is a fun and interactive tool that lets you explore these ideas! It’s like a virtual experiment where you can shoot a pretend object (called a projectile) and see how it moves.

This simulation is based on basic physics rules about motion. Don’t worry if you’re new to physics—we’ll explain everything step by step in simple terms. You don’t need any prior knowledge to get started. The goal is to help you understand how things move when thrown, using pictures, graphs, and easy explanations.

Getting Started: How to Use the Simulation#

The simulation is easy to use, even if it’s your first time. Imagine you’re playing with a toy launcher. You set how fast and at what angle to shoot, then watch it go!

Basic Buttons

Start: Click this to launch the projectile and see it fly.
Pause: Click this to stop the motion temporarily, like freezing time, so you can look closely at what’s happening.
Reset: Click this to start over. It clears everything and puts all settings back to the beginning.

Things You Can Change

You can adjust a few settings to see how they affect the projectile’s path:

Initial Speed (called \(v_0\)): This is how fast the projectile starts moving when you launch it. Think of it as how hard you throw a ball.
Launch Angle (called \(θ\)): This is the direction you aim, measured from flat ground (horizontal). For example, 0° is straight ahead, and 90° is straight up.
Gravity (called \(g\)): This is the pull of the Earth that makes things fall down. You can change it to see what happens on other planets!

Seeing the Data

The simulation shows graphs that plot what’s happening over time. You can pick what to show on the bottom (x-axis) and side (y-axis) of the graph. Options include:

Time (how long since launch),
Position in the horizontal direction (x-position),
Position in the vertical direction (y-position),
Speed in the horizontal direction (\(v_x\)),
Speed in the vertical direction (\(v_y\)).

These graphs help you visualize the motion, like watching a movie of the numbers!

Hint

Try different angles to see how far the projectile goes. For launching and landing on flat ground, test angles like 30°, 45°, and 60°. You’ll discover that 45° often gives the farthest distance! Watch the path—it looks like a curve called a parabola.

The Physics Behind It: Simple Explanations and Equations#

Physics is all about understanding motion. In this simulation, we focus on “projectile motion,” which is what happens when something is thrown and only gravity affects it (we ignore things like wind).

We assume two simple things:

No air resistance (nothing slows it down except gravity).
Gravity pulls straight down at a constant rate.

The cool part is that the motion can be split into two parts: side-to-side (horizontal) and up-and-down (vertical). They don’t affect each other!

Starting with Basics: What is Speed, Velocity, and Acceleration?

Speed is how fast something is going (like 10 meters per second).
Velocity is speed with direction (like 10 meters per second to the right).
Acceleration is how much velocity changes over time (like speeding up by 2 meters per second every second).

Gravity’s acceleration (\(g\)) is usually about 9.8 m/s² downward. That means falling things speed up by 9.8 m/s every second.

Breaking Down the Initial Speed

When you launch at an angle, the initial speed \(v_0\) splits into two parts using basic trigonometry (like triangle math):

Horizontal part: \(v_{0x} = v_0 \times \cos(θ)\) (stays the same forever, since nothing changes it).
Vertical part: \(v_{0y} = v_0 \times \sin(θ)\) (changes because of gravity).

Fundamental Equations of Motion

These come from simple algebra. If something moves with constant speed, distance = speed × time.

For constant acceleration, we use these basic equations (we’ll derive them step by step):

Velocity at any time: \(v = u + a t\) (final velocity = initial velocity + acceleration × time).
Position at any time: \(s = u t + \frac{1}{2} a t^2\) (distance = initial velocity × time + half acceleration × time squared).
Another one: \(v^2 = u^2 + 2 a s\) (final velocity squared = initial velocity squared + 2 × acceleration × distance).

Here, \(u\) is initial velocity, \(v\) is final velocity, \(a\) is acceleration, \(s\) is distance, \(t\) is time.

Applying to Horizontal Motion (No Acceleration, a_x = 0)

Velocity: \(v_x = v_{0x}\) (it’s constant, so just the starting value).
Position: Plug into the equation with a=0: \(x = v_{0x} t\) (assuming it starts at x=0).

That’s it—simple straight-line motion at constant speed.

Applying to Vertical Motion (Acceleration a_y = -g, downward)

Velocity: \(v_y = v_{0y} - g t\) (starts at \(v_{0y}\), subtracts \(g\) every second).
Position: \(y = v_{0y} t - \frac{1}{2} g t^2\) (assuming it starts at y=0).

How Do We Derive These Equations? (Algebra Only)

Let’s derive the position equation for vertical motion step by step.

We know acceleration is constant, so velocity changes linearly: average velocity = (initial + final)/2.

From the velocity equation: final \(v_y = v_{0y} - g t\).

Average vertical velocity over time t: \(\frac{v_{0y} + (v_{0y} - g t)}{2} = v_{0y} - \frac{1}{2} g t\).

Then, distance y = average velocity × time = \((v_{0y} - \frac{1}{2} g t) \times t = v_{0y} t - \frac{1}{2} g t^2\).

See? It’s just algebra—multiplying and averaging!

The horizontal one is even simpler: average velocity is just \(v_{0x}\) (since it doesn’t change), so \(x = v_{0x} t\).

Important Calculations for the Trajectory (on Flat Ground)

Let’s calculate some key values using the equations above. We’ll show each one with step-by-step algebra.

Time to Reach the Highest Point#

This is where the vertical velocity becomes zero (\(v_y = 0\)).

Start with the vertical velocity equation:

\[v_y = v_{0y} - g t_{peak} = 0\]

Solve for \(t_{peak}\):

\[t_{peak} = \frac{v_{0y}}{g} = \frac{v_0 \sin(θ)}{g}\]

Highest Height (H)#

Plug \(t_{peak}\) into the vertical position equation:

\[H = v_{0y} t_{peak} - \frac{1}{2} g t_{peak}^2\]

Substitute \(t_{peak}\):

\[H = (v_0 \sin(θ)) \left( \frac{v_0 \sin(θ)}{g} \right) - \frac{1}{2} g \left( \frac{v_0 \sin(θ)}{g} \right)^2\]

Simplify step by step:

First term: \(\frac{v_0^2 \sin^2(θ)}{g}\)

Second term: \(\frac{1}{2} g \cdot \frac{v_0^2 \sin^2(θ)}{g^2} = \frac{v_0^2 \sin^2(θ)}{2g}\)

So:

\[H = \frac{v_0^2 \sin^2(θ)}{g} - \frac{v_0^2 \sin^2(θ)}{2g} = \frac{2 v_0^2 \sin^2(θ)}{2g} - \frac{v_0^2 \sin^2(θ)}{2g} = \frac{v_0^2 \sin^2(θ)}{2g}\]

Total Time in the Air (T)#

By symmetry (time down equals time up):

\[T = 2 t_{peak} = \frac{2 v_0 \sin(θ)}{g}\]

How Far It Goes (Range R)#

Plug T into the horizontal position equation:

\[R = v_{0x} T = (v_0 \cos(θ)) \left( \frac{2 v_0 \sin(θ)}{g} \right) = \frac{2 v_0^2 \cos(θ) \sin(θ)}{g}\]

Using the identity \(2 \sin(θ) \cos(θ) = \sin(2θ)\):

\[R = \frac{v_0^2 \sin(2θ)}{g}\]

Key Things to Notice#

The side-to-side motion is steady, like cruising in a car. The up-and-down is like dropping something, speeding up as it falls. Together, they make a curved path—a parabola! You can see this if you plot y against x. To get the equation, start from \(x = v_{0x} t\), so \(t = x / v_{0x}\). Plug into y:

\[y = (v_{0y}) \left( \frac{x}{v_{0x}} \right) - \frac{1}{2} g \left( \frac{x}{v_{0x}} \right)^2 = (\tan(θ)) x - \frac{g}{2 v_{0x}^2} x^2\]

Why Use This Simulation in Learning?#

This tool helps you learn physics by doing:

Split 2D motion into horizontal and vertical parts.
Use trig to break vectors (like velocity) into pieces.
Apply simple motion equations to real scenarios.
Read graphs to understand speed and position changes.

It’s great for beginners to see physics in action!

If Something Goes Wrong#

Note

If the projectile flies off-screen or acts weird, try smaller speeds or angles.

If the app freezes, hit Reset. Make sure your settings aren’t too extreme (like super high speed). If problems persist, refresh the page.