Maximum Height Calculator

Compute the maximum height, flight time, horizontal range, and landing velocity for a projectile launched with initial speed, angle, and elevation. Now with linear air resistance modelling, animation, and data export. Perfect for physics students, ballistics engineers, sports analysts, and curious minds.

m/s
°
m
m/s²
All inputs are validated. Values outside reasonable ranges will show a warning.
Enter positive values. Angle in degrees (0–90°). Standard gravity g = 9.81 m/s² on Earth.
? Cannon
?️ Cliff
⚽ Football
? Moon
? Vertical
? High arc
?️ Golf
⚾ Baseball
? Javelin
Privacy first: All computations run locally in your browser. No data is transmitted or stored.

Understanding Projectile Maximum Height

In classical mechanics, projectile motion describes the motion of an object launched into the air and subject only to the force of gravity (and air resistance, when neglected). The maximum height — often denoted H or ymax — is the highest vertical position reached by the projectile during its flight. This occurs at the instant when the vertical component of velocity becomes zero: vy = 0.

H = y0 + ½ · (v0·sin θ)² / g

where v0 is initial speed, θ is launch angle, g is gravitational acceleration, and y0 is the initial height.

This elegant formula emerges from the kinematic equations of motion under constant acceleration. The vertical motion is independent of the horizontal motion — a foundational principle of Galilean relativity first articulated by Galileo Galilei in the early 17th century. The maximum height is directly proportional to the square of the initial speed and the square of the sine of the launch angle, and inversely proportional to gravity.

Why Use This Interactive Calculator?

  • Visual Learning: Watch the trajectory evolve as you adjust speed, angle, and height. Instantly see how the peak shifts and how the parabolic shape changes.
  • Educational Support: Verify homework solutions, explore "what‑if" scenarios, and build intuition for projectile motion — an essential topic in high school and university physics.
  • Engineering & Ballistics: Quickly estimate projectile behavior for design, simulation, or field operations (e.g., artillery, water jets, sports analytics).
  • Research & Experimentation: Test hypotheses about range, impact velocity, and optimal launch angles without needing a physical setup.

The Physics Behind the Calculation

The motion is decomposed into horizontal and vertical components. Assuming no air resistance, the horizontal velocity vx remains constant throughout the flight, while the vertical velocity vy changes uniformly under gravity.

Step 1 — Velocity components:

vx = v0 · cos θ    (constant)
vy(t) = v0 · sin θ − g · t

Step 2 — Time to peak:

Set vy(t) = 0  ⇒  tpeak = (v0 · sin θ) / g

Step 3 — Maximum height:

y(tpeak) = y0 + (v0 · sin θ) · tpeak − ½ · g · tpeak² = y0 + (v0² · sin² θ) / (2g)

Step 4 — Total flight time:

Solve y(t) = 0 for t > 0:
ttotal = (v0 · sin θ + √((v0 · sin θ)² + 2g·y0)) / g

Step 5 — Horizontal range:

R = vx · ttotal = v0 · cos θ · ttotal

Step 6 — Impact velocity:

vimpact = √(vx² + vy(ttotal)²)
where vy(ttotal) = v0 · sin θ − g · ttotal

When linear air resistance is enabled, the equations become:

With drag coefficient k (1/s):

vx(t) = v0x · e−kt
vy(t) = (v0y + g/k) · e−kt − g/k
x(t) = (v0x/k) · (1 − e−kt)
y(t) = y0 + (v0y + g/k)/k · (1 − e−kt) − (g/k)·t

Maximum height and flight time are computed numerically (using Brent's method) for the drag case.

Key Insights & Common Misconceptions

  • The 45° myth: While 45° maximizes range for a flat surface (h₀ = 0), for elevated launches (h₀ > 0) the optimal angle is less than 45°. This calculator lets you explore that directly.
  • Independence of motion: Horizontal and vertical motions are completely independent. The horizontal motion does not affect the vertical motion, and vice versa.
  • Maximum height vs. range: Increasing the launch angle increases height but decreases range (for a fixed speed). There is a fundamental trade‑off.
  • Air resistance: This tool now includes a linear drag model. In reality, air resistance reduces both maximum height and range, especially at higher speeds. The linear model is a good approximation for small, dense spheres at moderate speeds.
  • Symmetry: For h₀ = 0 and no drag, the trajectory is symmetric: the time up equals the time down, and the impact speed equals the launch speed. For h₀ > 0, the descent takes longer and impact speed exceeds launch speed.

Real‑World Applications

  • Sports Science: Analyze the trajectory of a soccer ball, basketball shot, or golf drive. Optimize launch parameters for maximum distance or accuracy.
  • Artillery & Ballistics: Determine the required initial velocity and angle to hit a target at a given range and elevation.
  • Water Fountains & Irrigation: Design fountain nozzles or sprinkler systems to achieve desired water coverage and height.
  • Aerospace: Model the initial phase of rocket launches (before active guidance) or the descent of re‑entry vehicles.
  • Video Game Physics: Implement realistic projectile behavior in game engines for enhanced player experience.
Case Study: Optimizing a Penalty Kick

A soccer player takes a penalty kick from the penalty spot (11 m from the goal line). The goal is 2.44 m high, and the player wants to shoot the ball over the wall (which is 9.15 m away) and under the crossbar. Using this calculator, the player can test different launch speeds and angles. For example, with v₀ = 25 m/s, θ = 25°, and h₀ = 0.2 m (the ball's height at contact), the ball reaches a maximum height of about 5.7 m at a horizontal distance of 12.8 m — clearing the wall and dipping into the goal. The tool's visual trajectory helps the player understand the flight path intuitively.

Step‑by‑Step Usage Guide

  1. Enter the initial speed (v₀) in meters per second.
  2. Set the launch angle (θ) in degrees (0–90°).
  3. Specify the initial height (h₀) above ground in meters.
  4. Adjust gravity (g) if needed — for example, use g = 1.62 m/s² for lunar conditions.
  5. Optionally, enable air resistance and set the drag coefficient k (typical values 0.01–0.5 1/s).
  6. Click Calculate & Draw to compute all parameters and visualize the trajectory.
  7. Use the preset examples to quickly explore classic scenarios.
  8. Click Play to see an animated ball follow the trajectory.
  9. Export the trajectory data as CSV or the graph as PNG.

Reference Table: Maximum Height for Common Scenarios

All values computed with the same physics model used in this tool (no drag).

Scenario v₀ (m/s) θ (°) h₀ (m) g (m/s²) Max Height (m) Range (m)
Golf drive 45 12 0.05 9.81 4.45 84.2
Baseball home run 42 30 1.0 9.81 23.5 161.8
Javelin throw 30 38 1.8 9.81 18.7 92.3
Long jump 9.5 22 0.5 9.81 1.21 8.24
Water fountain 12 55 0.3 9.81 5.52 13.9
Lunar landing 20 30 2.0 1.62 33.2 219.6

Frequently Asked Questions

The maximum height H is given by H = h₀ + (v₀² · sin² θ) / (2g), where h₀ is the initial height, v₀ is the initial speed, θ is the launch angle, and g is the gravitational acceleration. This formula assumes no air resistance.

For a projectile launched from ground level (h₀ = 0), the trajectory is perfectly symmetric: the time to reach the peak equals the time to fall back down. When h₀ > 0, the descent takes longer because the projectile has to fall further, so the peak occurs before the midpoint of the total flight time.

The maximum height is inversely proportional to gravity. On the Moon (g = 1.62 m/s²), a projectile reaches roughly 6 times the height it would on Earth for the same initial conditions. This is why lunar landers and astronauts can jump much higher on the Moon.

For a given initial speed, the maximum height is maximized when the launch angle is 90° (straight up). In this case, all the initial velocity is directed upward, and the projectile reaches its highest possible point. However, the range becomes zero because there is no horizontal motion.

Yes. Air resistance (drag) reduces both the maximum height and the range. The effect becomes more significant at higher speeds and for objects with larger cross‑sectional areas. This calculator now includes a linear drag model for more realistic simulations.

This calculator is designed for upward launches with angles between 0° and 90°. For downward launches (negative angles), the maximum height would simply be the initial height, and the trajectory would be different. We recommend using our Projectile Motion Calculator for full 360° support.

Built on classical mechanics — This tool is based on the kinematic equations of motion under uniform gravitational acceleration, as formalized by Galileo and Newton. The implementation follows standard physics textbooks (Halliday & Resnick, "Fundamentals of Physics"; Taylor, "Classical Mechanics"). The interactive graph uses HTML5 Canvas for real‑time rendering. Reviewed by the GetZenQuery tech team, last updated July 2026.

References: Wikipedia: Projectile Motion; Halliday, D., Resnick, R., & Walker, J. "Fundamentals of Physics" (10th ed.); Khan Academy: Physics.