Calculate 2D projectile motion trajectories, total flight time, horizontal range, maximum apex height, instantaneous velocity, and impact vectors with the free Projectile Motion Calculator.
Range: 163.15 m (535.3 ft) • Max Apex: 40.79 m • Apex Time: 2.89 s • Impact: 40.00 m/s @ -45.0°
535.3 ft
133.8 ft above ground
When Vy = 0 m/s
144.0 km/h
Constant velocity
Upward component
In classical Galilean mechanics, projectile motion describes the curved trajectory of an object launched into a uniform gravitational field. Because gravity acts strictly downward along the vertical axis, the motion decomposes into two completely independent orthogonal components:
In vacuum (\(a_x = 0\)), horizontal speed remains unchanged: \(v_x(t) = v_0\cos\theta\), causing horizontal displacement to grow linearly with time: \(x(t) = v_{0,x} t\).
Downward gravity (\(a_y = -g\)) decelerates upward ascent to a momentary zero-velocity apex (\(v_y = 0\)), then accelerates the projectile downward into free fall.
Eliminating time \(t\) from the parametric position equations produces the quadratic parabola: \(y(x) = h_0 + x\tan\theta - \frac{g x^2}{2 v_0^2 \cos^2\theta}\).
Complete summary of analytical kinematics formulas for flat and elevated 2D projectile motion:
| Kinematic Variable | Formula Equation | Physical Meaning |
|---|---|---|
| Initial Velocity Vector | $$v_{0,x} = v_0\cos\theta, \quad v_{0,y} = v_0\sin\theta$$ | Orthogonal decomposition of launch speed |
| Time to Apex (\(t_{\text{apex}}\)) | $$t_{\text{apex}} = \frac{v_0\sin\theta}{g}$$ | Duration until vertical velocity reaches zero |
| Max Peak Apex Height (\(H_{\text{max}}\)) | $$H_{\text{max}} = h_0 + \frac{(v_0\sin\theta)^2}{2g}$$ | Highest vertical elevation above ground |
| Total Flight Duration (\(T\)) | $$T = \frac{v_{0,y} + \sqrt{v_{0,y}^2 + 2 g h_0}}{g}$$ | Full hangtime until ground impact |
| Horizontal Range (\(R\)) | $$R = v_{0,x} \times T$$ | Total horizontal ground distance traveled |
Enter muzzle or release speed in \(\text{m/s}\), \(\text{km/h}\), \(\text{mph}\), \(\text{ft/s}\), or \(\text{knots}\).
Adjust launch angle from \(0^\circ\) to \(90^\circ\) and specify initial elevation above landing ground.
Drag the time slider to inspect exact \((x, y)\) coordinates, instantaneous velocity vectors, and trajectory angles during mid-flight.
Inspect total range, maximum apex height, flight time, and step-by-step KaTeX mathematical substitutions.
Why horizontal velocity never affects vertical fall time:
A bullet fired horizontally from a rifle and a bullet dropped from the exact same height at the same instant hit the ground simultaneously (in vacuum). Because gravity acts solely along the \(y\)-axis (\(\vec{g} = -g\hat{j}\)), horizontal velocity (\(v_x\)) has zero acceleration component along the vertical axis (\(\vec{a} \cdot \hat{i} = 0\)).
Inspect coordinates \((x(t), y(t))\) and velocity magnitude at any arbitrary mid-flight timestamp.
Solves both flat ground (\(h_0 = 0\)) and elevated cliff releases (\(h_0 > 0\)) using full quadratic roots.
Toggle between Earth, Moon (\(1.62\text{ m/s}^2\)), Mars (\(3.72\text{ m/s}^2\)), and Jupiter.
Calculates terminal landing speed and ground impact angle relative to the horizontal.
Renders clear algebraic substitution and quadratic kinematics solutions in real time.
Executes instantly on client device without server latency or data collection.
Prevents sign confusion and quadratic formula arithmetic mistakes in physics homework and laboratory problem sets.
Determines exact release angles and entry speeds for basketball three-pointers, golf trajectories, and soccer free kicks.
How atmospheric resistance reshapes the classical parabola:
In atmospheric conditions, aerodynamic drag (\(F_d = \frac{1}{2}\rho v^2 C_d A\)) decelerates projectiles proportionally to the square of velocity. This distorts the symmetrical parabola into an asymmetrical Tartaglia trajectory, where the descent is significantly steeper and shorter than the ascent, and downward terminal velocity limits maximum impact speed.
Optimizing trajectory kinematics in athletic competition:
Steeper hoop entry creates wider effective basket rim cross-section
Initial speed \(25-30\text{ m/s}\) with Magnus spin over defensive wall
High exit velocity with \(2,200\text{ RPM}\) backspin for maximum carry
Finding the exact firing angles to strike a target at distance \(d \le R_{\text{max}}\):
To hit a target on level ground at horizontal distance \(d\), rearrange the range equation to solve for the launch angle:
If \(\frac{gd}{v_0^2} > 1\), the target lies beyond maximum range (\(d > R_{\text{max}}\)) and cannot be reached without increasing launch velocity \(v_0\).
How gravity scaling impacts flight time and horizontal carry across the Solar System:
\(g = 1.62\text{ m/s}^2\) with zero vacuum air resistance
\(g = 3.72\text{ m/s}^2\) with ultra-thin \(0.006\text{ atm}\) atmosphere
\(g = 24.79\text{ m/s}^2\) severely compresses parabolic apex
When flat-Earth projectile motion transitions into orbital mechanics:
Standard 2D projectile kinematics assumes a flat ground plane with uniform downward gravity (\(g = \text{const}\)). For suborbital sounding rockets and ICBMs traveling thousands of kilometers, the Earth's spherical curvature and inverse-square central gravity (\(g(r) = \frac{GM}{r^2}\)) bend the flat parabola into an eccentric Keplerian ellipse with Earth's center of mass at one focal point.
Verifying that trajectory heights clear indoor structures and ceiling beams:
The instantaneous vertical trajectory height \(y(x)\) at any horizontal distance \(x\) along the path is given by the Cartesian trajectory formula:
Indoor sports designers and drone pilots use this equation to verify that basketball arcs or tennis lobs will not collide with overhead structural trusses.
How work-energy principles govern speed and apex energy transformation:
Total mechanical energy remains constant: \(E = \frac{1}{2}m v^2 + mgy = \text{constant}\). Final ground impact speed depends strictly on launch speed and height: \(|v_f| = \sqrt{v_0^2 + 2 g h_0}\), independent of launch angle.
At peak apex, vertical kinetic energy drops to zero, but total kinetic energy is never zero (\(E_{k,\text{apex}} = \frac{1}{2}m v_{0,x}^2\)), while potential energy peaks at \(U_{\text{apex}} = mg H_{\text{max}}\).
Kinematics when objects are thrown downward from elevated heights:
When tossing an object downward below the horizontal plane (\(\theta < 0^\circ\)), initial vertical velocity is negative (\(v_{0,y} = -v_0\sin|\theta|\)). There is no upward ascent phase, maximum height equals initial height (\(H_{\text{max}} = h_0\)), and flight time is abbreviated:
Engineering aerial mortar burst delays for symmetrical radial star explosions:
Commercial pyrotechnic display shells must detonate precisely at peak apex (\(t_{\text{apex}} = \frac{v_0\sin\theta}{g}\)) when vertical velocity is \(0\text{ m/s}\). Detonating at apex prevents downward trajectory distortion, creating a perfect spherical break while maintaining NFPA 1123 spectator safety fallout setback distances.
Comprehensive answers to common questions about 2D projectile motion equations, maximum apex height calculations, flight time formulas, and instantaneous velocity vectors.
