Maximum Height Calculator - Projectile Motion
Enter a launch speed, launch angle and starting height to find the maximum height a projectile reaches at the top of its arc. The calculator also shows the time to reach that peak, the horizontal range at landing, and the total time of flight. Switch between metric and imperial units or change the gravitational body, and the results update instantly.
What is the maximum height of a projectile?
When any object is launched at an angle, gravity continuously decelerates the vertical component of its velocity. The maximum height is the point where the vertical velocity momentarily reaches zero before the object begins to fall back. At that exact instant, the projectile is at its highest point. The horizontal velocity is unaffected by gravity (ignoring air resistance), so the object keeps moving forward throughout. The peak height depends only on three things: the launch speed, the launch angle, and the gravitational acceleration at the location.
The maximum height formula explained
The vertical velocity at launch is v_y = v_0 times sin(alpha), where v_0 is the launch speed and alpha is the angle above horizontal. As the projectile climbs, gravity subtracts g (about 9.807 m/s^2 on Earth) from v_y every second. The object stops rising when v_y reaches zero, which takes t_peak = v_y / g seconds. Substituting into the kinematic equation gives the height above the launch point: delta_h = v_y^2 / (2g). If the launch point is already at height h_0, the maximum height above the ground is h_max = h_0 + v_y^2 / (2g). For a 90 degree launch this simplifies to h_max = h_0 + v_0^2 / (2g). For 45 degrees it becomes h_max = h_0 + v_0^2 / (4g), exactly half the 90-degree value.
How to use this calculator
Choose metric or imperial units at the top, then enter the launch speed, the angle above horizontal (0 to 90 degrees), and the height of the launch point above the landing surface (enter 0 for ground-level launches). You can also switch the gravitational body to Moon or Mars to see how the lower gravity changes the trajectory. The results update instantly and include the maximum height, height gained above the launch point, time to reach the peak, total flight time, horizontal range at landing, and impact speed. The worked steps panel shows every calculation in detail using your actual numbers.
Gravity on other bodies
Gravitational acceleration varies by location. On Earth it is 9.807 m/s^2, on the Moon it is only 1.622 m/s^2 (about one-sixth), and on Mars it is 3.721 m/s^2 (about 38% of Earth). A lower g means the same launch makes a projectile rise higher and fly much farther. For example, a ball thrown at 20 m/s at 45 degrees reaches about 10.2 m on Earth, 61.9 m on the Moon, and 26.9 m on Mars. These comparisons are a useful way to build intuition about gravity as a physical constant.
Effect of launch angle on trajectory
| Angle (deg) | Max height (relative) | Range (relative) | Notes |
|---|---|---|---|
| 0 | 0.00 | 0.00 | Horizontal launch - no height gain |
| 15 | 0.07 | 0.50 | Low, flat arc |
| 30 | 0.25 | 0.87 | Half maximum height, good range |
| 45 | 0.50 | 1.00 | Maximum range on flat ground |
| 60 | 0.75 | 0.87 | Same range as 30 deg, more height |
| 75 | 0.93 | 0.50 | High arc, low range |
| 90 | 1.00 | 0.00 | Vertical launch - maximum height |
Relative maximum height and range for a fixed launch speed, compared to a 45 degree baseline (equal unit assumed).
Frequently asked questions
What launch angle gives the maximum height?
90 degrees (straight up) always gives the greatest possible height for a given launch speed, because all of the velocity is directed vertically. However, that produces zero horizontal range. For the greatest range on flat ground, 45 degrees is optimal, and it produces exactly half the height of a vertical launch.
Does a higher launch speed always mean more height?
Yes, for the same angle. Maximum height scales with the square of the vertical component of velocity: h = v_y^2 / (2g). Doubling the launch speed at the same angle quadruples the height gained above the launch point.
Does air resistance affect the maximum height?
Yes, significantly in real life. This calculator uses the standard projectile model which ignores air resistance. With drag, a real projectile reaches a lower maximum height and shorter range than these results show. For dense, slow objects at short distances the difference is small, but for lightweight or fast-moving objects it can be substantial.
Why is the time to peak not exactly half the total flight time?
When the launch height equals the landing height (h_0 = 0), the time to the peak is exactly half the flight time by symmetry. But when h_0 is greater than zero, the projectile falls farther on the way down than it climbed on the way up, so the descent takes longer and the total flight time is more than twice the time to the peak.
How do I find the horizontal range if I only know the maximum height?
You need at least two of the three original inputs (speed, angle, initial height) to compute the range independently. Maximum height alone is not enough to determine range, because many different speed-angle combinations can produce the same peak height with different ranges. Use this calculator with the original launch parameters to get the range alongside the height.