Maximum Height Calculator works out the peak of a projectile’s flight from its launch speed, angle and starting height. It also shows time to the peak, total flight time and range.
Measure the angle up from level ground. Pick 90° for a throw straight up.
Gravity and Units
Step-by-Step Working
- Known values in SI units: v₀ = 20 m/s, θ = 60°, h₀ = 1.5 m, g = 9.80665 m/s²
- Vertical launch speed: vy = v₀ sin θ = 20 × sin(60°) = 17.32 m/s
- Height gained: vy² ÷ 2g = (17.32)² ÷ (2 × 9.80665) = 15.3 m
- Time to the peak: vy ÷ g = 17.32 ÷ 9.80665 = 1.77 s
- Result: maximum height = h₀ + gain = 1.5 + 15.3 = 16.8 m
Height at Other Angles
| Launch Angle | Maximum Height | Range to Ground |
|---|---|---|
| 15° | 2.87 m | 24.97 m |
| 30° | 6.6 m | 37.75 m |
| 45° | 11.7 m | 42.24 m |
| 60° | 16.8 m | 36.17 m |
| 75° | 20.53 m | 20.79 m |
| 90° | 21.89 m | None |
Trajectory Graph
Maximum Height Calculator for Projectile Motion
The Maximum Height Calculator is a physics tool for objects that are thrown, kicked or launched. It gives the highest point of the flight above the ground. It does not predict how tall a person will grow, which is a different kind of height calculator.
Calculate From has four settings. Launch Speed and Angle is the usual one. Velocity Components takes the horizontal and vertical parts of the launch velocity. Range and Angle works back from how far the object travelled, and Target Height and Angle finds the launch speed needed to reach a chosen peak.
Every setting also takes a launch height, since most throws start above the ground. The default is 20 m/s at 60° from 1.5 m, which peaks at 16.8 m after 1.77 s. The cards add the total time in the air, the range, and the height the same speed would reach if thrown straight up.
Maximum Height Formula
A projectile stops rising when gravity has removed all of its upward speed. Only the vertical part of the launch velocity matters, which is the launch speed v0 times the sine of the angle θ. The maximum height H adds the climb to the launch height h0.
$$H = h_0 + \frac{(v_0 \sin\theta)^2}{2 g}$$
This is the result OpenStax University Physics reaches for any projectile with no air resistance. The time to reach the peak comes from the same vertical speed, and the first card of the Maximum Height Calculator reports it.
$$t_{peak} = \frac{v_0 \sin\theta}{g}$$
When the range R on level ground is known in place of the speed, the climb follows from the range and the angle alone. Gravity cancels out of this form, so it holds on any planet. The range here means the distance back to launch level.
$$H = h_0 + \frac{R \tan\theta}{4}$$
Turning the first formula around gives the launch speed needed for a target height. The Maximum Height Calculator uses this line in the Target Height and Angle setting, where the answer is a speed and not a height.
$$v_0 = \frac{\sqrt{2 g (H – h_0)}}{\sin\theta}$$
Maximum Height by Launch Angle
The table keeps the launch speed at 20 m/s from ground level and changes only the angle. The same table appears inside the Maximum Height Calculator for whatever speed is entered.
| Launch angle | Maximum height | Time to the peak | Range |
|---|---|---|---|
| 15° | 1.37 m | 0.53 s | 20.39 m |
| 30° | 5.1 m | 1.02 s | 35.32 m |
| 45° | 10.2 m | 1.44 s | 40.79 m |
| 60° | 15.3 m | 1.77 s | 35.32 m |
| 75° | 19.03 m | 1.97 s | 20.39 m |
| 90° | 20.39 m | 2.04 s | None |
Height keeps rising with the angle and is greatest at 90°, straight up. Range behaves differently. It peaks at 45° and is the same for pairs of angles that add to 90°, such as 30° and 60°. The steeper angle of each pair goes three times as high for the same distance.
A 60° launch reaches 75% of the straight-up height, because the share equals the sine of the angle squared. The fourth card of the Maximum Height Calculator shows this share for any entry. A level launch at 0° never rises above its start, and that case belongs to the Horizontal Projectile Motion Calculator.
Maximum Height by Launch Speed
Speed matters more than angle. The climb grows with the square of the vertical speed, so doubling the launch speed gives four times the height. The table is for a throw straight up from the ground on Earth.
| Launch speed | Maximum height | Time to the peak |
|---|---|---|
| 5 m/s | 1.27 m | 0.51 s |
| 10 m/s | 5.1 m | 1.02 s |
| 20 m/s | 20.39 m | 2.04 s |
| 30 m/s | 45.89 m | 3.06 s |
| 40 m/s | 81.58 m | 4.08 s |
Weaker gravity has the same effect as more speed. The default launch that peaks at 16.8 m on Earth reaches 42.04 m on Mars and 95.25 m on the Moon. On Jupiter it manages only 7.99 m. The gravity menu in the Maximum Height Calculator holds these presets.
Worked Maximum Height Problems
The OpenStax fireworks example is a direct check. A shell leaves at 70.0 m/s and 75° and bursts at its highest point. The textbook finds 233 m, reached after 6.90 s and 125 m away. The Maximum Height Calculator returns 233.1 m, 6.89 s and 124.92 m, with the small gap coming from g taken as 9.80665 in place of 9.8.
| Setting | Known values | Result |
|---|---|---|
| Launch Speed and Angle | 20 m/s at 60° from 1.5 m | 16.8 m maximum height |
| Launch Speed and Angle | 15 m/s straight up from 1 m | 12.47 m maximum height |
| Velocity Components | 12 m/s across, 16 m/s up, from 1.5 m | 14.55 m maximum height |
| Range and Angle | 40 m range at 60° from 1.5 m | 18.82 m maximum height |
| Target Height and Angle | 30 m peak at 60° from 1.5 m | 27.3 m/s launch speed |
The components row shows that the 12 m/s of sideways speed plays no part. Only the 16 m/s upward sets the peak, and the horizontal part decides where the peak happens. The launch works out to 20 m/s at 53.13°.
The target row depends strongly on the angle. Reaching a 30 m peak from 1.5 m takes 27.3 m/s at 60°, but 33.44 m/s at 45° and only 23.64 m/s straight up. A steeper launch wastes less of its speed on sideways motion.
Questions About Maximum Height
What angle gives the maximum height?
A launch of 90°, straight up, gives the greatest height for any speed. All of the launch speed then works against gravity. At 20 m/s from the ground that is 20.39 m, against 10.2 m at 45° and 5.1 m at 30°.
The angle for the longest range is a separate question. From the ground it is 45°, and from a raised start it is slightly lower. For 20 m/s from 1.5 m the best angle is 43.98°, which the fourth card reports.
Does the launch height change the maximum height?
It adds to it directly. The climb above the launch point depends only on the vertical speed and gravity, and the launch height sits underneath that. A 15.3 m climb from a 1.5 m start peaks at 16.8 m above the ground.
The launch height also lengthens the fall. From the ground, the time down equals the time up. From 1.5 m the default throw takes 1.77 s to rise and 1.85 s to come down, and it lands at 20.72 m/s, a little faster than it left.
How does air resistance affect maximum height?
Air resistance lowers the peak, because drag acts with gravity while the object is rising. The Maximum Height Calculator leaves it out, so its result is an upper limit. The error is small for dense, slow objects and large for light or fast ones.
Once the object passes its highest point, the fall from the peak is ordinary free fall, which the Free Fall Calculator covers on its own. A drop with drag included is handled by the Free Fall with Air Resistance Calculator.
What is the difference between maximum height and range?
Maximum height is how far up the object gets, and range is how far along the ground it lands. They respond to the angle in opposite ways above 45°. A steeper launch gains height and loses range.
On level ground the two are tied by the angle, since the height equals the range times tan θ divided by 4. At 45° the height is exactly a quarter of the range. A launch from a ramp with different heights at each end is the job of the Car Jump Distance Calculator.