Maximum Height of a Projectile Calculator

Enter the initial velocity and the angle of the launch of an object in projectile motion (assuming no air resistance) to calculate the maximum height of the projectile. This calculator can also evaluate the initial velocity or launch angle, given the height and the other variable.

• All Physics Calculators
• Free Fall Calculator
• Relative Velocity Calculator
• Launch Velocity Calculator

Launch Velocity Formula

This calculator estimates launch velocity from horizontal displacement and maximum height under ideal projectile-motion conditions. The relationship comes from finding the time required for the projectile to rise to its highest point and then dividing the horizontal distance by that rise time.

t = \sqrt{\frac{2H}{g}}

V = \frac{X}{t}

V = X \sqrt{\frac{g}{2H}}

In this equation, V is the calculated launch velocity, X is the horizontal change in position, H is the maximum height above the launch point, and g is gravitational acceleration. For most Earth-based problems, use 9.81 m/s² or 32.174 ft/s².

Rearranged Forms

If you already know the launch velocity and want to solve for another variable, these equivalent forms are helpful:

X = V \sqrt{\frac{2H}{g}}

H = \frac{gX^2}{2V^2}

How to Calculate Launch Velocity

1. Measure the horizontal change in position.
2. Measure the projectile’s maximum height relative to the launch point.
3. Select a consistent unit system so distance and velocity convert correctly.
4. Apply the formula to determine the launch velocity.

Because the equation depends on a square root, the maximum height must be greater than zero. Very small height values can produce very large velocity estimates, so accurate height measurement matters.

Example

If the horizontal change in position is 200 m and the maximum height is 40 m, the velocity is:

V = X \sqrt{\frac{g}{2H}}

V = 200 \sqrt{\frac{9.81}{2 \times 40}}

V \approx 70.04 \text{ m/s}

What the Inputs Mean

• Change in x-direction: the horizontal displacement used by the calculator.
• Maximum height: the highest vertical position reached above the launch point, not above sea level or some unrelated reference.
• Launch velocity: the speed computed from the relationship between horizontal travel and rise height.

Assumptions and Limitations

• The calculation assumes ideal projectile motion with no air resistance.
• Gravity is treated as constant for the full motion.
• The maximum height must be measured from the same launch reference used for the horizontal displacement.
• This relationship is most appropriate when the horizontal displacement and peak height describe the same motion interval.
• If your problem includes drag, wind, changing elevation, or additional thrust, a more complete projectile model is required.

Interpretation Tips

• If the horizontal displacement doubles while height stays the same, the calculated velocity doubles.
• If the maximum height increases while horizontal displacement stays the same, the calculated velocity decreases.
• The result is more sensitive to errors in horizontal distance than equal percentage errors in height.
• Unit conversions matter: a small mistake between feet and meters can significantly change the answer.

Common Questions

Does this calculator include air resistance?

No. It assumes ideal motion, so drag, lift, and wind effects are ignored.

Should maximum height be measured from the ground?

Only if the launch point is at ground level. In general, height should be measured from the actual launch point to the apex.

Can I use imperial units?

Yes. Feet, yards, miles, feet per second, and miles per hour are acceptable as long as the values are entered consistently.

Why does a larger height reduce the calculated velocity in this formula?

A larger peak height means the projectile spends more time rising. For the same horizontal displacement, more rise time means less horizontal speed is needed to cover that distance.