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

Horizontal velocity

14.14

m/s

vₓ — constant throughout the flight

Vertical velocity

14.14

m/s

vᵧ — at the moment of impact

Impact velocity

m/s

Speed when hitting the ground

Projectile Trajectory

g = 9.80665 m/s²

Fundamentals & Explanation

Galileo’s Principle of Independence

A projectile subject only to gravity moves in a plane, but Galileo showed that this two-dimensional motion is the superposition of two independent straight-line motions that share the same clock and do not affect each other:

Horizontal axis — uniform motion. With no horizontal force (air ignored), the velocity vₓ = v₀·cos θ stays constant for the entire flight.
Vertical axis — uniform acceleration. Gravity imposes a constant downward acceleration, so vᵧ = v₀·sin θ − g·t changes linearly with time.

x(t) = v_0\cos\theta\;t \qquad y(t) = h_0 + v_0\sin\theta\;t - \tfrac{1}{2}g\,t^2

These two parametric equations hold the entire kinematics of the problem: eliminating time between them yields the familiar parabola y(x). The calculator uses g = 9.80665 m/s² (standard surface gravity).

The Trajectory’s Key Quantities

The three quantities shown in the results panel follow directly from the position equations:

Flight time

t = \dfrac{v_0\sin\theta + \sqrt{(v_0\sin\theta)^2 + 2gh_0}}{g}

Max height

y_{\max} = h_0 + \dfrac{(v_0\sin\theta)^2}{2g}

Range

X = v_0\cos\theta \cdot t

The flight time is the positive root of y(t) = 0 (the quadratic formula), and remains valid when launching from a height h₀ > 0.
The maximum height is reached at the apex, where vᵧ = 0, at the instant t = v₀·sin θ / g.
The range is the horizontal position at landing: since vₓ is constant, just multiply it by the total flight time.

Mind the range formula. The compact expression X = v₀²·sin(2θ) / g found in many textbooks only holds when launch and landing are at the same level (h₀ = 0). When h₀ > 0, the projectile falls below its launch point and you must use the general form X = v₀·cos θ · t with the full flight time — which is exactly what this simulator computes.

The Optimal Angle and the 45° Symmetry

Launching from ground level (h₀ = 0), the range is proportional to sin(2θ), which peaks (at 1) when 2θ = 90°. That is why the maximum range occurs at 45°. The same symmetry means that two complementary angles (θ and 90° − θ, e.g. 30° and 60°) yield exactly the same range — one with a flat, fast trajectory, the other high and slow.

When launching from a height (h₀ > 0), the projectile has extra time to fall, so it pays to flatten the shot a bit: the optimal angle drops below 45° and equals

\theta_{\text{opt}} = \arctan\!\left(\dfrac{v_0}{\sqrt{v_0^2 + 2gh_0}}\right)

which reduces to 45° when h₀ = 0, recovering the classic case.

Impact Velocity and Energy Conservation

During the flight, the horizontal component vₓ never changes, while the vertical one vᵧ vanishes at the apex and then grows downward. At landing, the components are

v_x = v_0\cos\theta \qquad |v_y| = \sqrt{(v_0\sin\theta)^2 + 2gh_0}

and the impact speed is their vector composition, which simplifies remarkably:

v_f = \sqrt{v_x^2 + v_y^2} = \sqrt{v_0^2 + 2gh_0}

This result is independent of the angle: it is exactly mechanical energy conservation in a vacuum (½mv² + mgy = const). All the potential energy of the initial height h₀ turns into kinetic energy, so a projectile launched higher always hits faster, regardless of the firing direction.

Gravity on Other Worlds

Gravity g inversely scales both range and height. With the same speed and angle, a shot travels much farther where g is smaller:

Body	g (m/s²)	Effect
Earth	9.80665	reference
Moon	1.62	range ~6×
Mars	3.72	range ~2.6×
Jupiter	24.79	abrupt fall

Model Limits and Assumptions

No air resistance

The model assumes motion in a vacuum. In reality, drag (proportional to v² at high speeds) shortens the range and makes the trajectory asymmetric — the descending branch is steeper. That is the topic of a separate projectile-with-drag calculator.

Flat Earth and constant g

Earth’s curvature and the variation of g with altitude are ignored. For very long-range projectiles (ballistic missiles), the direction of “down” changes and elliptical orbital mechanics is needed.

No Coriolis effect

Earth’s rotation deflects long-duration projectiles sideways (relevant in heavy artillery or very long-distance shooting). Here the reference frame is treated as inertial.

Quantity	Symbol	SI unit
Initial velocity	v₀	meter/second (m/s)
Launch angle	θ	radian (rad) / degree
Initial height	h₀	meter (m)
Gravity	g	m/s²
Flight time	t	second (s)

References and Resources

[1] OpenStax (2023). University Physics, Vol. 1, §4.3 “Projectile Motion” (open access).
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.), Ch. 4. Cengage.
[3] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.), Ch. 4. Wiley.

Projectile Motion Simulator

Input parameters

Results

Projectile Trajectory

Fundamentals & Explanation

Galileo’s Principle of Independence

The Trajectory’s Key Quantities

The Optimal Angle and the 45° Symmetry

Impact Velocity and Energy Conservation

Gravity on Other Worlds

Model Limits and Assumptions

No air resistance

Flat Earth and constant g

No Coriolis effect

References and Resources

You may also like:

Galileo’s Principle of Independence

The Trajectory’s Key Quantities

The Optimal Angle and the 45° Symmetry

Impact Velocity and Energy Conservation

Gravity on Other Worlds

Model Limits and Assumptions

No air resistance

Flat Earth and constant g

No Coriolis effect

References and Resources