Analyze horizontal projectile motion instantly. Compute time of flight, horizontal range, impact velocity, and landing angle from initial speed and height. Visualize the parabolic trajectory, velocity components, and key kinematic parameters on an interactive canvas.
Horizontal projectile motion describes the motion of an object launched horizontally from an elevated position, subject only to the force of gravity. This classic physics problem is fundamental to understanding kinematics in two dimensions. The object's horizontal velocity remains constant (neglecting air resistance), while its vertical velocity increases uniformly under gravitational acceleration. The resulting path is a parabola – a shape that appears throughout nature, from water fountains to ballistic trajectories.
The motion is governed by two independent sets of equations:
Horizontal: x(t) = v₀ · t | Vertical: y(t) = h − ½ · g · t²
where v₀ is the initial horizontal speed, h is the launch height, g is the acceleration due to gravity, and t is time.
The concept of projectile motion has been studied since the time of Galileo Galilei (1564–1642), who first demonstrated that the trajectory of a projectile is a parabola. Galileo's insight — that horizontal and vertical motions are independent — laid the foundation for classical mechanics and later influenced Newton's laws of motion. Today, projectile motion analysis is essential in fields ranging from ballistics and aerospace engineering to sports science and video game physics.
In horizontal projectile motion, the initial velocity has no vertical component. This simplifies the analysis: the time of flight depends only on the initial height and gravity, not on the horizontal speed. The horizontal range scales linearly with the initial speed, making it easy to predict outcomes for different launch conditions. This independence is a hallmark of Newtonian physics and a powerful teaching tool for understanding vector components.
Starting from the kinematic equations with zero initial vertical velocity, we derive the key quantities:
Time of flight (T): From y(T) = 0 → 0 = h − ½·g·T² → T = √(2h/g)
Horizontal range (R): R = v₀ · T → R = v₀ · √(2h/g)
Impact velocity (v): vx = v₀ (constant), vy = g·T → v = √(v₀² + (gT)²)
Landing angle (θ): θ = arctan(vy / vx) → θ = arctan(gT / v₀)
These equations show that the time of flight is independent of the horizontal speed — a surprising and counterintuitive result that is central to understanding projectile motion. The range, however, is directly proportional to the initial speed and the square root of the height.
The following table shows example results verified against analytical solutions. You can reproduce these by selecting the corresponding preset examples.
| Scenario | v₀ (m/s) | h (m) | g (m/s²) | T (s) | R (m) | v (m/s) | θ (°) |
|---|---|---|---|---|---|---|---|
| Earth Standard | 10 | 20 | 9.8 | 2.02 | 20.20 | 22.18 | 63.4 |
| High Speed | 30 | 50 | 9.8 | 3.19 | 95.83 | 43.47 | 55.4 |
| Low Height | 5 | 5 | 9.8 | 1.01 | 5.05 | 11.10 | 63.4 |
| Moon | 15 | 30 | 1.62 | 6.09 | 91.33 | 17.95 | 33.1 |
| Mars | 12 | 25 | 3.71 | 3.67 | 44.05 | 18.15 | 48.7 |
A physics class is conducting a water balloon launch experiment from the roof of a school building. The launch point is 15 m above the ground. Students launch the balloon horizontally with a speed of 8 m/s. Using our calculator:
The trajectory graph clearly shows the parabolic path, helping students connect the mathematical equations with the physical reality. The class can then vary the launch speed and observe how the range changes, reinforcing the concept of independent horizontal and vertical motions.
One of the most profound insights in classical mechanics is the independence of perpendicular motions. In horizontal projectile motion, the vertical motion (governed by gravity) is completely independent of the horizontal motion (which has no acceleration). This means that:
The calculator visualizes this independence by showing the trajectory and allowing you to change the horizontal speed while observing that the time of flight remains constant. This interactive exploration deepens conceptual understanding far beyond rote memorization of formulas.