Maximum Height Projectile Motion
Calculate the maximum vertical height reached by a projectile using initial velocity, launch angle and initial height. Free online physics calculator with interactive charts for students and engineers.
About This Calculator
The Maximum Height Projectile Motion Calculator determines the highest vertical position a launched object reaches during its flight. It uses the standard kinematic equations for projectile motion under ideal conditions (no air resistance, uniform gravity) to compute the apex height, time to reach the apex, and vertical velocity component.
When a projectile is launched at an angle θ with initial velocity v, the vertical component of velocity is vy = v × sin(θ). Gravity acts downward at g = 9.80665 m/s², reducing the upward velocity until it reaches zero at the peak. The maximum height formula is hmax = h + vy² / (2g) = h + (v × sin(θ))² / (2g), where h is the initial launch height. This calculator computes the exact apex for any launch angle between 0 and 90 degrees, including elevated launches from an initial height.
This calculator is ideal for physics students, educators, sports scientists tracking ball trajectories, and engineers analyzing projectile systems. Results include the maximum height, time to reach maximum height, and the vertical component of velocity, all displayed with interactive bar and pie charts plus a detailed breakdown table.
Regional Notes
India (IN): Uses metric units (m/s, m, degrees) standard in Indian physics curricula (CBSE, ICSE, state boards).
United States (US): Uses metric units (m/s, m, degrees) as standard in physics education; the gravitational constant 9.80665 m/s² applies universally.
United Kingdom (UK): Uses metric units (m/s, m, degrees) as standard in A-level and GCSE physics curricula.
Frequently Asked Questions
How do you find the maximum height of a projectile?
The maximum height of a projectile is found using the formula hmax = h + (v × sin(θ))² / (2g), where v is the initial velocity, θ is the launch angle, h is the initial height, and g is the acceleration due to gravity (9.80665 m/s²). At the maximum height, the vertical velocity becomes zero.
What is the formula for maximum height in projectile motion?
For a projectile launched from ground level (h = 0), the maximum height formula is hmax = (v × sin(θ))² / (2g). When launched from an initial height h, the formula becomes hmax = h + (v × sin(θ))² / (2g), where v is the initial speed, θ is the launch angle, and g is 9.80665 m/s².
How do you calculate the time to reach maximum height?
The time to reach maximum height is given by t = v × sin(θ) / g, where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (9.80665 m/s²). This is derived from the kinematic equation vy = v × sin(θ) − gt, setting vy = 0 at the apex.
What angle gives the maximum height for a projectile?
A launch angle of 90 degrees (straight up) gives the maximum height for a given initial velocity because sin(90°) = 1, maximizing the vertical component of velocity. However, this results in zero horizontal range since all velocity is directed upward.
Does the mass of the projectile affect maximum height?
No, the mass of the projectile does not affect the maximum height in ideal projectile motion. The maximum height depends only on the initial velocity, launch angle, and initial height. The acceleration due to gravity is the same for all objects regardless of mass (in the absence of air resistance).
Does air resistance affect the maximum height?
Yes, air resistance (drag) reduces the maximum height a projectile can reach. This calculator uses the ideal projectile motion model which ignores air resistance, providing accurate results for most introductory physics problems. For real-world applications with significant air resistance, the actual height will be lower.
What is the difference between maximum height and range in projectile motion?
Maximum height is the highest vertical position reached during projectile motion, while range is the total horizontal distance traveled. They are related but distinct: a 45-degree angle maximizes range on level ground, while a 90-degree angle maximizes height but gives zero range.