Physics Intelligence

Projectile Motion
Professional Simulator

Visualize the physics of flight. Explore how gravity, launch angle, and velocity dictate the trajectory of objects in motion with absolute precision.

✓ Trajectory Graphing✓ SUVAT Equation Solving✓ Real-time Vector Decomposition

Launch Parameters

Initial Velocity (v₀)20 m/s

1 m/s100 m/s

Launch Angle (θ)45°

0°45°90°

Initial Height (h)0 m

0 m100 m

Trajectory Simulation

Flight Data

Max Height

10.19m

Range

40.77m

Time

2.88s

Velocity (Horizontal)14.14 m/s

Velocity (Vertical Initial)14.14 m/s

Projectile motion occurs when an object is launched into the air and moves only under the influence of gravity (ignoring air resistance). This motion follows a parabolic path, which is the result of constant horizontal velocity combined with constant vertical acceleration.

In classical mechanics, these two dimensions are perfectly independent. High-authority physics models like this simulator treat the horizontal and vertical components as separate mathematical entities to predict exactly where and when an object will land.

The Insider’s Guide to Trajectory Control

Predicting flight isn't just about plugging in numbers; it's about understanding Opportunity Cost in physics—where energy used for height is energy lost for range.

1. The "45° Myth" in Real-World Scenarios

Standard textbooks teach that 45° yields the maximum range. While true on flat ground in a vacuum, it changes drastically if you launch from a height (like a cliff).

Strategic Application: If your launch point is higher than your landing target, your optimal angle for range actually decreases (often to 35-40°). If launching to a higher target, you must launch at a steeper angle.

2. High vs. Low Trajectories (Complementary Angles)

In physics, two launch angles that sum to 90° (e.g., 30° and 60°) will land at the exact same spot.

Insider Insight: The 60° launch (High Trajectory) provides more "hang time," which is useful in sports like football or soccer to allow teammates to reach the target. The 30° launch (Low Trajectory) reaches the target faster, providing less time for air resistance to disrupt the path.

The Mathematics of Ballistics

The simulator utilizes the fundamental SUVAT Equations for constant acceleration. We decompose the initial velocity vector $v_0$ into components using trigonometry.

Horizontal (x)

v_x = v_0 \cos(\theta)

x(t) = v_x \cdot t

Horizontal velocity is constant ( $a_x = 0$ ).

Vertical (y)

v_y = v_0 \sin(\theta) - gt

y(t) = h + v_{y0}t - \frac{1}{2}gt^2

Gravity accumulates over time ( $g \approx 9.81 m/s^2$ ).

Maximum Range Formula

For launch and landing at the same height:

R = \frac{v^2 \sin(2\theta)}{g}

Scenario	Launch Angle	Comparative Outcome
Flat Range Max	45°	Standard Maximum Distance
Cliff-Top Launch	35°	Optimal Range from Height
"Hail Mary" Pass	60°+	High Hang-Time / Short Range

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Does mass affect the trajectory?

Experimentally, No. In our high-precision model (which ignores air resistance), mass is not a variable. Galileo famously proved that all objects fall at the same rate regardless of weight. A bowling ball and a tennis ball launched at the same angle will follow identical paths.

What happens if I launch at 90°?

The horizontal velocity ( $\cos(90^\circ)$ ) becomes zero. The projectile moves perfectly vertically, reaches its peak, and falls back precisely where it started. This yields maximum "Max Height" but zero range.

How does gravity vary by location?

We use Earth's standard gravity ( $g = 9.81 m/s^2$ ). On the Moon, gravity is $1/6^{th}$ as strong, meaning the same launch would achieve 6x the range and height!

Is air resistance included?

This is a Kinematic Model, which assumes a vacuum. In real-world ballistics, "drag" acts in opposition to motion, creating a non-parabolic path that falls shorter and steeper than predicted here.

What is the "Zenith"?

The Zenith is the peak of the trajectory. At this point, the vertical velocity is exactly zero before the object begins its descent.

Trajectory

The curved path followed by a projectile under the action of gravity.

SUVAT

Acronym for the variables of motion: Displacement (s), Initial Velocity (u), Final Velocity (v), Acceleration (a), and Time (t).

Vector Decomposition

The process of breaking a diagonal force or velocity into its pure horizontal and vertical components.

Hang Time

The total duration an object remains in the air before impacting the ground.

Fact-Checked by the CalculatorsCentral Physics Lab. SUVAT integrations verified against classical mechanics benchmarks.

Last Updated: January 2026

This simulator assumes zero air resistance and a constant gravitational field. While precise for theoretical physics, real-world ballistic calculations should account for air density, wind, and the Coriolis effect for long-range targets.

Projectile Motion Professional Simulator