Projectile Motion Simulator

Kinematics — calculate range, maximum height, and trajectory of a projectile.

Input parameters

Initial Velocity

Speed at which the projectile is launched

0100

Launch Angle

Angle with respect to horizontal

090

Initial Height

Height from which the projectile is launched

050

Results

Flight time

2.884

Maximum range

40.79

Maximum height

10.2

Projectile Trajectory

g = 9.80665 m/s²

Fundamentals & Explanation

Position Equations

x(t) = v_0 \cos(\theta)\, t

y(t) = h_0 + v_0 \sin(\theta)\, t - \frac{1}{2}g\,t^2

Parametric equations of motion. $v_0$ is the initial velocity, $\theta$ the launch angle, and $h_0$ the initial height.

Maximum Range (X)

X_{\max} = \frac{v_0^2 \sin(2\theta)}{g}

Total horizontal distance covered. This simplified formula assumes the projectile launches and lands at the same elevation ( $h_0 = 0$ ).

Maximum Height (Y)

Y_{\max} = h_0 + \frac{(v_0 \sin\theta)^2}{2g}

The apex of the parabola, where the vertical velocity reaches zero ( $v_y = 0$ ) before the object begins to fall.

Flight Time

t = \frac{v_0 \sin\theta + \sqrt{(v_0 \sin\theta)^2 + 2g h_0}}{g}

Derived by solving the quadratic equation for $y(t) = 0$ and taking the positive root.

1. Galileo's Principle of Superposition

Galileo Galilei's genius was proving that the two-dimensional motion of a projectile is simply the superposition of two completely independent one-dimensional motions:

Horizontal AxisUniform Rectilinear Motion (URM). With no horizontal forces acting on the object (ignoring air resistance), the velocity $v_x$ remains constant throughout the flight.
Vertical AxisUniformly Accelerated Motion (UAM). The object experiences a constant downward acceleration due to Earth's gravity ( $g \approx 9.81\text{ m/s}^2$ ).

2. The Geometry of 45 Degrees

If you launch an object from ground level ( $h_0 = 0$ ), the theoretical maximum range is always achieved at an angle of 45°. Mathematically, this happens because the function $\sin(2\theta)$ reaches its maximum value (1) when $2\theta = 90^\circ$ .

However, if you launch the object from an elevated position ( $h_0 > 0$ ), the projectile spends more time in the air falling below its launch point. In this case, the optimal angle to maximize horizontal range is slightly less than 45°.

3. Reference Gravitational Acceleration (g)

This calculator uses standard Earth gravity by default. If you wish to simulate projectile motion on other celestial bodies, here are the approximate values:

Celestial Body	Gravity (g)	Trajectory Effect
Earth (Standard)	9.80665 m/s²	—
Moon	1.62 m/s²	Range ~6 times longer
Mars	3.72 m/s²	Range ~2.6 times longer
Jupiter	24.79 m/s²	Steep and immediate drop

4. Assumptions and Model Limitations

①No Aerodynamic Drag. The model assumes the projectile moves in a vacuum. In reality, air friction slows the object down (drag force is proportional to the square of the velocity). At high speeds (e.g., a bullet), the actual trajectory is much shorter and asymmetrical.
②Flat Earth and Constant Gravity. Earth's curvature is ignored. For ultra-long-range projectiles (like ballistic missiles), the direction of "down" changes constantly, requiring elliptical orbital mechanics.
③Coriolis Effect. Earth's rotation slightly deflects long-duration projectiles (a noticeable factor in heavy artillery or extreme sniping), which is omitted in this idealized calculator.

University References

OpenStax (2023) — University Physics Vol. 1: Projectile Motion (Open Access)
Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th Ed.). Cengage.
Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th Ed.). Wiley.

Projectile Motion Simulator

Input parameters

Results

Projectile Trajectory

Fundamentals & Explanation

1. Galileo's Principle of Superposition

2. The Geometry of 45 Degrees

3. Reference Gravitational Acceleration (g)

4. Assumptions and Model Limitations

University References

You may also like:

1. Galileo's Principle of Superposition

2. The Geometry of 45 Degrees

3. Reference Gravitational Acceleration (g)

4. Assumptions and Model Limitations

University References