Maximum Displacement Calculator: Formula & Examples

This calculator works out the maximum displacement of a projectile—the farthest vertical height it reaches above its launch point—using the launch speed, launch angle, and gravity. You enter those three values and get back the peak height in meters (or feet), plus the time it takes to get there. The formula is the standard kinematic equation for vertical motion under constant gravity.

Formula

The maximum displacement (often called maximum height) of a projectile is given by:

h_max = (v₀² · sin²θ) / (2g)

Where:

• h_max = maximum vertical displacement (m or ft)
• v₀ = initial launch speed (m/s or ft/s)
• θ = launch angle above the horizontal (degrees)
• g = acceleration due to gravity (9.81 m/s² on Earth, 32.17 ft/s² in imperial)

The time to reach that height is:

t_up = (v₀ · sinθ) / g

This assumes no air resistance, a flat launch and landing surface, and that the projectile starts at ground level (height zero). If you launch from a cliff or a raised platform, add that initial height to the result.

Worked Example

Using the default values from the calculator above: launch speed v₀ = 20 m/s, angle θ = 45°, gravity g = 9.81 m/s².

First, find the vertical component of the launch speed:

v_y = v₀ · sinθ = 20 · sin(45°) = 20 · 0.7071 = 14.142 m/s

Now apply the maximum displacement formula:

h_max = (v_y²) / (2g) = (14.142²) / (2 · 9.81) = 200 / 19.62 = 10.19 m

Time to reach that height:

t_up = v_y / g = 14.142 / 9.81 = 1.44 s

The calculator shows 10.19 m and 1.44 s, matching these steps.

When to Use This Calculator

This is the tool for any physics problem that asks “how high does it go?”—a ball thrown upward, a rocket launched at an angle, a water jet from a fountain, or a projectile in a game. It is also useful for checking whether an object will clear a wall or obstacle of known height.

Keep in mind the result is the maximum vertical displacement above the launch point. If the projectile starts from a height (e.g., a cannon on a hill), add that starting height to the result to get the absolute height above ground.

For horizontal distance (range), use a different formula: R = (v₀² · sin2θ) / g. This calculator only handles the vertical peak.

Common Mistakes and How to Avoid Them

• Angle in radians vs. degrees: The formula uses sin of the angle. If your calculator is in radian mode, sin(45) gives 0.8509 instead of 0.7071, changing the result by about 20%. Always set your calculator to degrees, or convert: θ in radians = θ in degrees × π/180.
• Mixing units: If speed is in m/s, gravity must be in m/s² (9.81). If speed is in ft/s, gravity must be in ft/s² (32.17). Mixing them gives wildly wrong numbers.
• Forgetting the square: The speed is squared in the numerator. Doubling the launch speed quadruples the maximum height, not just doubles it.
• Using total flight time instead of time to peak: The time to maximum height is half the total flight time (for a flat surface). The calculator gives the time to the peak.

Limits of the Formula

The formula assumes ideal conditions: no air resistance, constant gravity, and a flat, horizontal launch and landing surface. In real life, air drag slows the projectile, so the actual maximum height is lower than the calculated value—especially for fast or lightweight objects like a ping-pong ball or a bullet. For dense, slow objects (a thrown rock), the error is small.

Gravity also varies slightly with altitude and latitude (about 9.78 to 9.83 m/s² on Earth’s surface), but for most purposes 9.81 m/s² is accurate enough.

The formula also assumes the launch and landing points are at the same height. If the projectile lands on a lower or higher surface, the maximum height is still the same, but the total flight time and range change.

How to Read the Result

The maximum displacement tells you the highest point the projectile reaches. If you are aiming over a wall, compare this value to the wall height. If the wall is 8 m tall and the projectile reaches 10.19 m, it clears the wall with 2.19 m to spare.

The time to max height is useful for timing when to trigger something at the peak (e.g., a camera shutter or a mid-air collision). Remember that the projectile takes the same amount of time to fall back down to the launch height, so total flight time is roughly twice the time to peak.

Frequently Asked Questions

What is the difference between maximum displacement and maximum height?

They are the same thing in this context. Maximum displacement refers to the greatest vertical distance from the launch point, which is the maximum height. Some textbooks use “displacement” to emphasize it is a vector quantity (with direction), but the magnitude is the height.

Does the maximum height depend on the mass of the projectile?

In the ideal formula, no. Mass cancels out because both the initial kinetic energy and the gravitational potential energy depend on mass. A heavier object and a lighter object with the same launch speed and angle reach the same maximum height (ignoring air resistance). With air resistance, lighter objects are affected more and may not reach the same height.

What angle gives the maximum height for a given speed?

90 degrees (straight up) gives the highest possible maximum height, because all the speed goes into the vertical component. At 45 degrees, the range is maximized, but the height is only half of the straight-up value. The calculator will show that: at 45°, h = v²/(4g), while at 90°, h = v²/(2g).

Can I use this for a projectile launched from a moving vehicle?

Yes, if you use the velocity relative to the ground. For example, if you throw a ball upward from a car moving at 30 m/s, the vertical launch speed is what matters for the maximum height. The horizontal speed does not affect the vertical motion (again, ignoring air resistance). So you would enter the vertical component of the throw speed, not the car’s speed.

Why does the calculator show a different result than my textbook?

Check the units. If your textbook uses feet and pounds, enter the speed in ft/s and gravity as 32.17 ft/s². Also check whether the textbook includes air resistance or a different launch height. The calculator uses the ideal formula with no air resistance and launch from ground level.

How do I find the maximum displacement if I only know the total flight time?

If you know the total flight time T and the launch angle, you can find the initial vertical speed: v_y = g·T/2. Then use h = v_y²/(2g). Alternatively, the maximum height is simply h = g·T²/8. This works for a flat surface with no air resistance.

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in v | Launch speed | m/s | 20 | 0
in a | Launch angle | deg | 45 | 0
in g | Gravity | m/s² | 9.81 | 0
out h | Maximum displacement | m | (v^2 * (sind(a))^2) / (2 * g) | 2
out t | Time to max height | s | v * sind(a) / g | 2