91.7431
m
22.9358
m
4.3248
s
21.2132
m/s
21.2132
m/s
30
m/s
45
°
91.7431
m
22.9358
m
4.3248
s
21.2132
m/s
21.2132
m/s
30
m/s
45
°
The Projectile Motion Calculator analyzes the two-dimensional trajectory of an object launched at an angle to the horizontal. Projectile motion is a cornerstone topic in classical mechanics, combining horizontal constant-velocity motion with vertical free-fall acceleration to produce the characteristic parabolic trajectory.
When you throw a ball, kick a soccer ball, hit a golf shot, or fire a cannon, the resulting motion follows the laws of projectile physics. The horizontal component of velocity remains constant (ignoring air resistance), while the vertical component changes linearly due to gravitational acceleration. Together, these produce a parabolic arc described by a set of elegant mathematical equations.
This calculator takes the initial speed, launch angle, and optionally the initial height above ground level. It computes the horizontal range (how far the projectile travels), the maximum height reached, the total time of flight, the velocity components, and the impact velocity and angle when the projectile returns to ground level. For launches from an elevated position (h₀ > 0), the calculator uses the full quadratic solution to account for the extra fall distance.
One of the most famous results in projectile motion is that the maximum range on level ground occurs at a launch angle of 45°. This is because the range formula R = v₀²sin(2θ)/g is maximized when sin(2θ) = 1, which occurs at θ = 45°. However, when launching from an elevated position, the optimal angle is less than 45°, and when launching uphill, it is greater than 45°.
Projectile motion analysis is used extensively in sports science (optimizing throw and kick angles), military ballistics (artillery trajectory computation), civil engineering (water fountain design, debris trajectory analysis), space exploration (landing trajectory planning), and entertainment (fireworks choreography, stunt coordination). Even video game physics engines implement these equations to simulate realistic object trajectories.
The calculator ignores air resistance, which is a reasonable approximation for dense, compact objects at moderate speeds. For high-speed projectiles, long-range shots, or light objects like shuttlecocks, aerodynamic drag significantly alters the trajectory, reducing range and maximum height. The gravitational acceleration can be adjusted in advanced settings to model projectile motion on other planets or celestial bodies.
The projectile velocity is decomposed into horizontal and vertical components:
$$v_{0x} = v_0 \cos\theta, \quad v_{0y} = v_0 \sin\theta$$
Time of flight (from ground level, h₀ = 0):
$$T = \frac{2 v_0 \sin\theta}{g}$$
When launched from height h₀ > 0:
$$T = \frac{v_{0y} + \sqrt{v_{0y}^2 + 2gh_0}}{g}$$
Horizontal range:
$$R = v_{0x} \cdot T$$
For flat ground (h₀ = 0), this simplifies to the classic formula:
$$R = \frac{v_0^2 \sin(2\theta)}{g}$$
Maximum height above ground:
$$H = h_0 + \frac{v_{0y}^2}{2g} = h_0 + \frac{v_0^2 \sin^2\theta}{2g}$$
Impact velocity:
$$v_{y,\text{impact}} = v_{0y} - gT$$
$$v_{\text{impact}} = \sqrt{v_{0x}^2 + v_{y,\text{impact}}^2}$$
Impact angle:
$$\alpha = \arctan\left(\frac{|v_{y,\text{impact}}|}{v_{0x}}\right)$$
The range tells you how far the projectile lands from the launch point. Maximum height is the peak altitude above the ground. Time of flight is the total airborne duration. The horizontal velocity component remains constant throughout the flight, while the vertical component changes direction at the peak. The impact velocity is typically equal to the launch speed for flat-ground launches (by energy conservation) but directed downward. The impact angle is measured from the horizontal.
Inputs
Results
A soccer ball kicked at 25 m/s at 30° travels 55.2 m with a max height of 8 m and stays airborne for 2.55 seconds.
Inputs
Results
A ball launched at 20 m/s at 45° from a 30-meter cliff travels 57.1 m horizontally, reaching a peak of 40.2 m above ground before hitting at 28.1 m/s.
On level ground with no air resistance, the range is R = v₀²sin(2θ)/g. The sine function reaches its maximum value of 1 when 2θ = 90°, giving θ = 45°. At this angle, the horizontal and vertical velocity components are equal, providing the best balance between height (time in air) and horizontal speed. With air resistance or uneven ground, the optimal angle differs.
In the idealized model (no air resistance), mass does not affect the trajectory at all. The path depends only on initial velocity, launch angle, and gravitational acceleration. In reality, heavier objects are less affected by air resistance relative to their weight, so they follow the idealized trajectory more closely than lighter objects.
Air resistance (drag) reduces both the range and maximum height. The trajectory becomes asymmetric—the descending portion is steeper than the ascending portion. The optimal launch angle decreases below 45° (typically 30–40° for real baseballs and golf balls). Drag force depends on speed, cross-sectional area, and drag coefficient, making exact calculations require numerical methods.
Yes. Set the initial height (h₀) in the advanced settings to the height above the landing level. The calculator uses the full quadratic solution for time of flight, correctly accounting for the extra fall distance. The maximum height output includes h₀.
At 0° (horizontal launch from ground level), the projectile immediately hits the ground with zero flight time and range. At 90° (straight up), the range is zero and the projectile goes to its maximum height and falls straight back down. The calculator accepts angles between these extremes (exclusive) for meaningful two-dimensional motion.
This calculator provides the idealized parabolic trajectory used as a first approximation in ballistics. Real artillery calculations must account for air resistance (drag), wind, the Coriolis effect (Earth’s rotation), air density changes with altitude, and the projectile’s spin. Modern fire-control computers use numerical integration of the equations of motion including all these factors.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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