Compute peak height, time to apex, flight time, and range — using velocity & angle or components
Powered by Toolraxy · Physics & kinematics calculator
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Faiq Ur Rahman is a web designer, digital product developer, and founder of Toolraxy, a growing platform of web-based calculators and utility tools. He specializes in building structured, user-friendly tools focused on health, finance, productivity, and everyday problem-solving.
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When an object is launched into the air, understanding how high it will travel is essential for everything from sports analysis to engineering design. The maximum height of a projectile depends on launch speed, angle, initial elevation, and the gravitational pull acting on it. This maximum height calculator handles all these variables, giving you a complete picture of projectile motion with just a few inputs.
Physics students studying kinematics will find this tool invaluable for checking homework problems. Engineers designing trajectories for rockets, projectiles, or even sports equipment can quickly explore how changes in launch conditions affect performance. The calculator computes not just peak height, but also the time to reach that height, total flight duration, and horizontal range — all while letting you adjust gravity for different planets or custom values.
Toolraxy built this projectile motion calculator to make complex physics calculations accessible. Everything runs locally in your browser, so your data stays private while you explore the fascinating world of ballistic motion.
Enter the launch velocity in your preferred unit — m/s, km/h, mph, or ft/s.
Set the launch angle between 0 and 90 degrees above the horizontal.
Adjust the initial height if launching from an elevated position (like a cliff or platform).
Choose a gravity preset for Earth, Moon, Mars, or Jupiter, or enter a custom value.
Watch the components update automatically as you type — horizontal and vertical velocities sync with your main inputs.
Review the results: maximum height, time to peak, total flight time, and horizontal range all appear instantly.
Projectile motion follows the laws of kinematics, with gravity acting vertically while horizontal motion remains constant. The calculator determines several key parameters using standard physics equations:
Maximum Height: H = h₀ + (vᵧ²) / (2g)
The peak height equals the initial launch height plus the vertical distance traveled upward. The term vᵧ²/(2g) represents the height gained from the initial upward velocity, where vᵧ is the vertical velocity component and g is gravitational acceleration.
Time to Apex: t_peak = vᵧ / g
This is how many seconds the projectile takes to reach its maximum height. At this instant, the vertical velocity becomes zero before the object begins falling back down.
Total Flight Time: t_total = (vᵧ + √(vᵧ² + 2gh₀)) / g
When launching from ground level (h₀ = 0), the total flight time is simply twice the time to peak. For elevated launches, the calculator solves the quadratic equation h₀ + vᵧt – ½gt² = 0 to find when the projectile returns to ground level.
Horizontal Range: R = vₓ × t_total
The range equals the horizontal velocity component multiplied by the total flight time.
Component Relationships: vₓ = v₀·cos(θ) and vᵧ = v₀·sin(θ)
These equations break the launch velocity into horizontal and vertical components based on the launch angle θ.
A soccer ball is kicked from ground level at 25 meters per second with a launch angle of 30 degrees. Let’s find how high it goes and how far it travels.
Step 1: Identify the inputs — v₀ = 25 m/s, θ = 30°, h₀ = 0 m, g = 9.81 m/s².
Step 2: Break velocity into components — vₓ = 25 × cos(30°) = 21.65 m/s, vᵧ = 25 × sin(30°) = 12.5 m/s.
Step 3: Calculate time to peak — t_peak = 12.5 / 9.81 = 1.27 seconds.
Step 4: Find maximum height — H = 0 + (12.5²) / (2 × 9.81) = 156.25 / 19.62 = 7.96 meters.
Step 5: Determine total flight time — since h₀ = 0, t_total = 2 × 1.27 = 2.54 seconds.
Step 6: Compute range — R = 21.65 × 2.54 = 54.99 meters.
The ball reaches a peak height of about 8 meters, stays airborne for roughly 2.5 seconds, and travels nearly 55 meters downfield. This example illustrates why soccer players kick at moderate angles — too low and the ball stays near the ground, too high and it sacrifices distance for height.
The maximum height formula is H = h₀ + (v₀² × sin²θ) / (2g), where v₀ is launch speed, θ is launch angle, h₀ is initial height, and g is gravitational acceleration. This combines the vertical component of velocity with kinematic relationships to find the peak of the trajectory.
The launch angle determines what fraction of the total velocity goes upward. A 90-degree angle sends all velocity vertically, producing the maximum possible height for a given speed. A 45-degree angle splits velocity equally between horizontal and vertical, balancing height and range. Shallower angles produce lower heights but longer ranges.
For a launch from ground level, total flight time is exactly twice the time to peak because the upward and downward journeys are symmetric. When launching from an elevated position, total flight time exceeds twice the peak time because the projectile has further to fall after reaching the top.
Gravity acts as the force pulling the projectile downward. Stronger gravity reduces maximum height for a given launch because it decelerates the upward motion more quickly and accelerates the fall more rapidly. On the Moon, where gravity is about one-sixth of Earth’s, the same launch would reach about six times the height.
Yes, the calculator solves the quadratic equation to find when the projectile returns to ground level from any initial height. It handles positive initial heights (launching upward from an elevated position) and can also handle negative values if you want to analyze launches into a hole or below ground level.
At 45 degrees, the product of horizontal and vertical velocity components is maximized for a given launch speed. This balances the competing effects of higher launch angle (more height and time) versus lower angle (more horizontal speed). The result is the maximum range on level ground, assuming no air resistance.
The calculations assume ideal projectile motion with no air resistance, constant gravity, and a flat Earth. Real-world projectiles experience drag, wind, varying gravity with altitude, and other effects that cause deviations. The results are excellent for educational purposes, approximations, and initial planning, but professional applications should account for additional factors.
The calculator accepts velocity in m/s, km/h, mph, and ft/s. Initial height and output distances appear in meters, feet, and kilometers. Gravity presets cover Earth, Moon, Mars, and Jupiter, with a custom option for any value. All conversions happen automatically.
A negative time value indicates the projectile is already past its peak when launched — this happens with negative vertical velocity (launching downward). The calculator handles this by showing the appropriate time values, though launches with downward components typically aren’t considered “maximum height” scenarios in the traditional sense.
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