Calculate a projectile's maximum height (the highest point) and time to the peak from launch speed and angle or from velocity components, with optional starting height and gravity.
Advanced options
How to use our Maximum Height Calculator (Projectile Motion)
- Pick a Mode: use Max height to find the peak from your launch, or Solve to find the speed needed to reach a target peak height.
- Enter Initial height h0 (m) (your starting height, measured from your chosen zero level) and Gravity g (m/s^2) (enter g as a positive number).
- Open Advanced options and choose Input form: Speed and angle (use Initial speed v0 (m/s) plus angle) or Components (use Initial horizontal speed v0x (m/s) and Initial vertical speed v0y (m/s)).
- If you use an angle, set Angle unit first (degrees or radians), then enter the angle in that same unit.
- If Mode is Solve, enter Target maximum height h_max target (m) (same reference level as h0), then choose Solve for = required v0 (needs an angle) or required v0y (does not need an angle).
- (Optional) Set Number display and Significant figures to match how your class wants rounding shown.
- Click Calculate.
- Sanity-check the result: if Initial vertical speed used v0y (m/s) is 0 or negative, then Time to reach maximum height (s) should be 0 and Maximum height h_max (m) should equal Initial height h0 (m).
Definitions
Projectile motion: Motion where the only acceleration is gravity straight down (no air resistance, no thrust) [3].
Maximum height h_max (m): The highest vertical position reached, measured from the same reference level as Initial height h0 (m).
Initial height h0 (m): Starting vertical position at time zero. It can be negative if your chosen zero level is above the launch point.
Vertical component v0y (m/s): The up-down part of the initial velocity (up is positive). At the highest point, the vertical velocity is 0 [1].
Horizontal component v0x (m/s): The left-right part of the initial velocity. In this no-air-resistance model, it does not change the maximum height [1].
Gravity g (m/s^2): The positive magnitude of the downward acceleration used in the equations.
Degrees vs radians: Two ways to measure angles. 180 degrees equals about 3.1416 radians, so the numbers look very different.
Time to peak t_to_peak (s): Time from launch until the vertical velocity reaches 0 (the top of the path) [1].
Significant figures: A rounding rule for how many meaningful digits to show (example: 12.34 with 3 significant figures becomes 12.3).
Common mistakes and quick fixes
Mistake: Typing an angle in radians (like 0.8) while Angle unit is set to degrees, or typing degrees (like 45) while set to radians.
Fix: Set Angle unit first, then re-enter the angle value in that unit.
Mistake: Entering Gravity g (m/s^2) as a negative number because gravity points downward.
Fix: Enter g as a positive magnitude (example: 9.80665). The downward direction is already built into the equations.
Mistake: Choosing Input form = Components but changing Initial speed v0 (m/s) or the angle and expecting the results to change.
Fix: In Components mode, only Initial horizontal speed v0x (m/s) and Initial vertical speed v0y (m/s) are used. Switch back to Speed and angle to use v0 and angle.
Mistake: Getting h_max = h0 and thinking the calculator is broken.
Fix: Check Initial vertical speed used v0y (m/s) . If v0y is 0 or negative, the projectile does not rise, so the highest point is the launch point and Time to reach maximum height (s) is 0.
Mistake: In Mode = Solve, choosing Solve for = required Initial speed v0 (m/s) with an angle that has no upward part (for example 0 degrees), then seeing N/A.
Fix: Use an angle with a positive upward component (sine greater than 0), or switch Solve for to required Initial vertical speed v0y (m/s) .
Mistake: Entering Target maximum height h_max target (m) using a different zero level than Initial height h0 (m) (for example, one measured from ground and one from the launch point).
Fix: Make sure h0 and the target height are measured from the same reference level, or the required speed/height will look wrong.
Limitations & Key Assumptions / Boundary Conditions
No air resistance (drag): Real objects usually reach a lower maximum height because air drag reduces the upward speed. This calculator can overestimate the peak.
Gravity is constant and straight down: The model uses one constant Gravity g (m/s^2) for the whole motion. At very high altitudes or long distances, g and direction can vary.
Not a powered flight model: This is for an object after release (no thrust). Rockets, drones, and jumping people do not follow these equations while powered.
Peak-at-launch rule when not thrown upward: If Initial vertical speed used v0y (m/s) is 0 or negative, the highest point is the launch point, so Time to reach maximum height (s) = 0 and Maximum height h_max (m) = Initial height h0 (m).
Solve mode feasibility depends on angle: When solving for required Initial speed v0 (m/s), angles with no upward component (sine less than or equal to 0) cannot reach a higher target peak, so the result is N/A.
Same reference level required: Initial height h0 (m) and Target maximum height h_max target (m) must use the same zero level (same "ground" reference) for Solve mode to make sense.
Numeric overflow: Extremely large inputs can produce non-finite values (Infinity or NaN). If you see N/A, reduce the size of the inputs to realistic values.
Methodology
This calculator uses constant-acceleration projectile motion: horizontal acceleration is 0 and vertical acceleration is -g (downward), with no air resistance [2]. The maximum height happens when the vertical velocity becomes 0 [1].
1) Get initial velocity components. In speed-and-angle mode, the calculator converts the angle to radians if needed, then computes v0x and v0y. In components mode, it uses your entered v0x and v0y directly.
v0x = v0 * cos(θ), v0y = v0 * sin(θ)
2) Time to peak. If the initial vertical speed is not upward, the peak is at launch, so time to peak is 0.
t_to_peak = max(0, v0y / g)
3) Maximum height. For an upward launch (v0y greater than 0), the vertical height gained until the vertical speed reaches 0 is v0y^2 divided by (2*g). Otherwise, the maximum height is the launch height.
h_max = h0 + (v0y^2) / (2 * g) (if v0y > 0; else h_max = h0)
delta_h = h_max - h0
4) Solve mode (match a target maximum height). A target below h0 is impossible because the launch point is already higher. For a target equal to h0, zero launch speed is a solution. For a higher target, calculate the required upward vertical speed.
v0y_required = sqrt(2 * g * (h_target - h0)), valid only when h_target >= h0
If solving for total launch speed, divide by sin(θ). If sin(θ) is not positive and the target is above h0, reaching the target is not possible in this model, so the calculator shows N/A.
v0_required = v0y_required / sin(θ)
Worked mini-example: v0 = 20 m/s, θ = 45 degrees, h0 = 0 m, g = 9.80665 m/s^2. Then v0y = 20 * sin(45 degrees) = 14.142 m/s, t_to_peak = 14.142/9.80665 = 1.442 s, and h_max = 0 + (14.142^2)/(2*9.80665) = 10.19 m.
Display and rounding: results are rounded to your chosen significant figures. Auto number display switches to scientific notation only for very large or very small nonzero values to keep results readable.