Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. In ideal projectile motion (without air resistance), the horizontal velocity remains constant while the vertical motion is affected by gravity, resulting in a parabolic path.

On level ground with no air resistance, the optimal launch angle for maximum horizontal range is exactly 45°. This gives the best balance between horizontal velocity and time of flight. However, if launching from an elevated position, the optimal angle becomes slightly less than 45°. For example, from a height of 10m with initial speed 20 m/s, the optimal angle is approximately 42°.

Air resistance (drag) opposes the motion of the projectile, reducing both its horizontal range and maximum height. In the real world, the optimal launch angle for maximum range is typically less than 45° (often around 30-40°) depending on the object's shape, size, and speed. Our simulator uses ideal conditions without air resistance for simplicity, which is accurate for dense, fast objects over short distances.

The key formulas used are:
• Horizontal velocity: v_x = v₀ × cos(θ) (constant)
• Initial vertical velocity: v_y0 = v₀ × sin(θ)
• Time of flight: t = (v_y0 + √(v_y0² + 2gh₀)) / g
• Maximum height: h_max = h₀ + v_y0² / (2g)
• Horizontal range: R = v_x × t
These assume constant gravity and no air resistance.

Gravity varies significantly across the solar system. Earth's gravity is 9.81 m/s², while the Moon's is only 1.62 m/s² (about 1/6th). Mars has 3.71 m/s², and Jupiter has a massive 24.79 m/s². Try switching between these presets in our simulator to see how dramatically the trajectory changes — on the Moon, a projectile can travel six times farther!

When you launch from an elevated position, the projectile has more time in the air before hitting the ground. This extra flight time allows the horizontal velocity to carry it further. Additionally, the projectile gains vertical speed as it falls below the launch height, increasing the impact speed. This is why launching from a hilltop or elevated platform significantly increases range.

Projectile motion principles apply to many fields including: sports (golf, baseball, basketball, soccer, javelin throw), military (ballistics, artillery, missile trajectories), engineering (water fountains, fireworks, demolition), space exploration (rocket launches, planetary landers), and even forensic science (crime scene reconstruction involving bullet trajectories).

The impact angle is the angle at which the projectile strikes the ground, measured from the horizontal. It is calculated using the horizontal and vertical velocity components at the moment of impact: θ_impact = arctan(|v_y| / v_x). The vertical velocity at impact is: v_y = -√(v_y0² + 2gh₀). For symmetric trajectories (h₀=0), the impact angle equals the launch angle.