Calculate projectile maximum apex height (H = h₀ + v₀²·sin²θ/2g), time to apex (t_apex = v₀·sinθ/g), horizontal range, flight time, and kinetic energy with the free Maximum Height Calculator.
43.19 ft • Time to Apex: 1.64 s • Total Range: 75.09 m • Total Flight Time: 3.28 s
Vertical v_y = 0
246.36 ft
Ground impact
82.6 km/h (51.3 mph)
E_k = 1/2 m v_x²
100.8 km/h
In classical Newtonian mechanics and kinematics, the maximum height (also known as the apex or vertical peak altitude, denoted as \(H\)) of a projectile is the highest vertical position reached along its curved parabolic path. At this instantaneous peak, the projectile's upward vertical motion ceases completely before gravitational acceleration pulls it downward:
At the exact peak of flight, the vertical velocity component decelerates to zero: \(v_y(t_{\text{apex}}) = v_0 \sin\theta - g t_{\text{apex}} = 0\).
In the absence of air resistance, the horizontal velocity component remains constant throughout the entire trajectory: \(v_x = v_0 \cos\theta\).
All vertical kinetic energy (\(\frac{1}{2} m v_{0y}^2\)) is converted into maximum gravitational potential energy (\(m g \Delta h\)).
Summary of the foundational kinematic equations used across physics, ballistics, and sports:
| Kinematic Metric | Mathematical Formula | SI Units & Description |
|---|---|---|
| Maximum Apex Height (\(H\)) | $$H = h_0 + \frac{v_0^2 \sin^2\theta}{2g}$$ | \(\text{meters (m)} \equiv \text{feet (ft)}\) |
| Time to Reach Apex (\(t_{\text{apex}}\)) | $$t_{\text{apex}} = \frac{v_0 \sin\theta}{g}$$ | \(\text{seconds (s)}\) |
| Horizontal Velocity at Apex (\(v_x\)) | $$v_x = v_0 \cos\theta$$ | \(\text{m/s} \equiv \text{ft/s}\) |
| Total Flight Time (\(T_{\text{total}}\)) | $$T = \frac{v_0 \sin\theta + \sqrt{(v_0 \sin\theta)^2 + 2 g h_0}}{g}$$ | \(\text{seconds (s)}\) |
| Total Horizontal Range (\(R\)) | $$R = v_0 \cos\theta \times T_{\text{total}}$$ | \(\text{meters (m)}\) |
| Inverse Velocity Solver (\(v_0\)) | $$v_0 = \frac{\sqrt{2g(H - h_0)}}{\sin\theta}$$ | \(\text{m/s}\) |
Choose between Forward Kinematics (Solve Height \(H\)), Inverse Velocity Solver (\(v_0\)), or Inverse Angle Solver (\(\theta\)).
Input launch velocity in \(\text{m/s}\), \(\text{km/h}\), \(\text{ft/s}\), or \(\text{mph}\) and angle in degrees or radians.
Input release elevation (\(h_0\)) and select gravity field (Earth, Moon, Mars, Jupiter, or Custom).
Inspect peak altitude, flight duration, apex speed, kinetic energy ratio, and step-by-step KaTeX mathematical derivations.
How angular trajectory trade-offs dictate vertical altitude versus horizontal distance:
Because \(\sin(90^\circ) = 1.0\), \(100\%\) of initial kinetic energy is directed into vertical ascent: \(H_{\text{max}} = h_0 + \frac{v_0^2}{2g}\).
Splits initial velocity equally between vertical and horizontal components (\(v_{0x} = v_{0y} = v_0 / \sqrt{2}\)), producing exactly \(50\%\) of vertical max height while maximizing horizontal range: \(R = \frac{v_0^2}{g}\).
Accounting for launch platforms, buildings, and human release heights:
When launching from an elevated platform of height \(h_0\), the time to reach the apex (\(t_{\text{apex}}\)) and the relative height above the launch point (\(\Delta H = \frac{v_0^2 \sin^2\theta}{2g}\)) remain identical to a ground-level launch. However, the total altitude above the ground is the direct sum \(H_{\text{total}} = h_0 + \Delta H\), and the falling phase after the apex lasts longer due to the extra descent distance.
Solve for apex height (\(H\)), required launch velocity (\(v_0\)), or launch angle (\(\theta\)) with automatic unit conversions.
Evaluate trajectories under Earth (\(9.807\text{ m/s}^2\)), Moon (\(1.62\text{ m/s}^2\)), Mars (\(3.71\text{ m/s}^2\)), or Jupiter (\(24.79\text{ m/s}^2\)).
Accommodates non-zero launch heights (\(h_0\)) with full quadratic descent time solutions.
Computes horizontal velocity and kinetic energy retention ratio at the parabolic apex.
Renders clear algebraic proofs with live numeric substitutions for physics coursework.
Runs instantly on any smartphone, tablet, or desktop with zero server lag and total calculation privacy.
Prevents common student errors when squaring trigonometric terms (\(\sin^2\theta\)) and converting between degrees and radians.
Eliminates tedious manual quadratic formula factoring when calculating flight time and range from elevated cliffs (\(h_0 > 0\)).
Comparing maximum height scaling across the solar system for a standard \(20\text{ m/s}\) throw at \(45^\circ\):
| Celestial Body | Gravity (\(g\)) | Apex Height (\(H\)) | Time to Apex | Relative Height Factor |
|---|---|---|---|---|
| Earth | \(9.81\text{ m/s}^2\) | \(10.19\text{ m}\) | \(1.44\text{ s}\) | \(1.00\times\) (Baseline) |
| Moon | \(1.62\text{ m/s}^2\) | \(61.73\text{ m}\) | \(8.73\text{ s}\) | \(6.05\times\text{ Higher}\) |
| Mars | \(3.71\text{ m/s}^2\) | \(26.95\text{ m}\) | \(3.81\text{ s}\) | \(2.64\times\text{ Higher}\) |
| Jupiter | \(24.79\text{ m/s}^2\) | \(4.03\text{ m}\) | \(0.57\text{ s}\) | \(0.40\times\text{ Lower}\) |
Why real-world projectiles fall short of ideal vacuum maximum height:
In real atmospheric conditions, quadratic air drag force (\(F_d = \frac{1}{2} \rho v^2 C_d A\)) continuously opposes the projectile's velocity vector. On high-speed objects like baseballs or golf balls, air resistance reduces peak apex height by \(15\%\text{ to }35\%\) compared to the ideal vacuum equation, while shifting the trajectory apex slightly forward along the flight path.
How initial kinetic energy partitions perfectly between horizontal motion and gravitational altitude:
The vertical kinetic energy converts entirely into potential energy: \(E_{p,\text{apex}} = m g (H - h_0) = E_{k,0} \sin^2\theta\).
Because horizontal velocity is invariant (\(v_x = v_0 \cos\theta\)), the projectile retains kinetic energy: \(E_{k,\text{apex}} = \frac{1}{2} m v_x^2 = E_{k,0} \cos^2\theta\).
How athletes adjust launch angle and velocity to control trajectory apex:
In basketball, shooters aim for a \(45^\circ\text{ to }55^\circ\) release angle (producing an apex altitude of approximately \(4.5\text{ to }5.0\text{ meters}\)) because a steeper descent angle widens the effective elliptical cross-section of the 18-inch rim, significantly increasing shooting percentage. In soccer, free-kick specialists launch at \(30^\circ\text{ to }38^\circ\) to clear the defensive wall while diving under the crossbar.
Calculating maximum altitude after rocket booster motor burnout:
For sounding rockets and ballistic missiles, powered ascent terminates at motor burnout altitude (\(y_{\text{bo}}\)) with burnout velocity (\(v_{\text{bo}}\)). The unpowered ballistic coast to peak apogee is governed directly by kinematics:
Deriving apex height perpendicular to an inclined mountain slope:
When firing a projectile at angle \(\theta\) along a hill inclined at angle \(\alpha\), the gravity vector resolves into components parallel (\(g \sin\alpha\)) and perpendicular (\(g \cos\alpha\)) to the slope. The maximum perpendicular height above the incline is:
The theoretical protective boundary for all launch angles fired at speed \(v_0\):
For any fixed initial speed \(v_0\), varying the launch angle \(\theta\) from \(0^\circ\) to \(90^\circ\) generates a family of intersecting parabolas. The mathematical envelope enclosing all possible trajectories defines the Parabola of Safety:
Any point located outside this bounding parabola is physically unreachable, defining the safety exclusion zone used in explosive quarry blasting and artillery defense planning.
How backspin and topspin alter apex altitude and parabolic curvature:
In golf driver shots and baseball pitches, rapid backspin creates high pressure below and low pressure above the ball, generating upward lift (\(\vec{F}_L = S(\vec{\omega} \times \vec{v})\)) that increases apex height by up to \(30\%\).
In tennis and table tennis groundstrokes, topspin drives airflow faster underneath the ball, generating downward lift that suppresses apex height and forces a sharp, rapid dive into the opponent's court.
Comprehensive answers to common questions about projectile maximum height formulas, time to peak apex, initial velocity calculations, and kinematics.
