Range Projectile Motion
Calculate the horizontal range, time of flight, and maximum height of a projectile using initial velocity, launch angle, and initial height. Free online physics calculator with interactive charts and formula breakdown for students and engineers.
About This Calculator
The Range Projectile Motion Calculator computes the horizontal range, time of flight, and maximum height of a projectile launched with a given initial velocity and angle. It uses the standard kinematic equations for projectile motion under ideal conditions (no air resistance, uniform gravity).
The range formula derives from the parabolic trajectory equations: the horizontal component of velocity remains constant at v × cos(θ), while the vertical component follows v × sin(θ) − gt under constant gravitational acceleration g = 9.80665 m/s². The range is the product of horizontal velocity and time of flight, giving R = 2v² × sin(θ) × cos(θ) / g = v² × sin(2θ) / g for ground-level launches. For elevated launches, the extended formula accounts for the additional fall time from height h.
This calculator is ideal for physics students, educators, ballistics engineers, and anyone studying kinematics. It supports both ground-level and elevated launch scenarios, with an optional initial height input. Results include the horizontal and vertical velocity components (Vx and Vy) in a detailed breakdown table, plus interactive bar and pie charts.
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 is consistently 9.80665 m/s² regardless of region.
United Kingdom (UK): Uses metric units (m/s, m, degrees) as standard in A-level and GCSE physics curricula.
Frequently Asked Questions
What is the formula for projectile range?
For a projectile launched from ground level, the horizontal range is R = v² × sin(2θ) / g, where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (9.80665 m/s²). For launches from a height, the formula is R = v × cos(θ) × (v × sin(θ) + √((v × sin(θ))² + 2gh)) / g.
What is the optimal angle for maximum projectile range?
The optimal launch angle for maximum range on level ground is 45 degrees. This is because sin(2θ) in the range formula reaches its maximum value of 1 when 2θ = 90°, so θ = 45°. This assumes no air resistance and a flat surface.
How do you calculate the time of flight of a projectile?
From ground level, time of flight t = 2v × sin(θ) / g. From an initial height h, t = (v × sin(θ) + √((v × sin(θ))² + 2gh)) / g. Time of flight depends only on the vertical component of velocity and the initial height.
How do you find the maximum height of a projectile?
Maximum height reached by a projectile launched from ground is hmax = (v × sin(θ))² / (2g). When launched from an initial height h, the total maximum height is h + (v × sin(θ))² / (2g).
Does air resistance affect projectile motion?
Yes, air resistance (drag) affects real-world projectile motion by reducing both range and maximum height. This calculator uses the ideal projectile motion model which ignores air resistance, making it accurate for most educational and introductory physics applications.
What is the difference between range and projectile motion?
Projectile motion is the entire parabolic trajectory of an object under gravity. The range is specifically the total horizontal distance traveled from launch point to landing point. This calculator focuses on computing the range, time of flight, and maximum height as key projectile motion parameters.
Can this calculator handle launches from a height?
Yes, this calculator supports both ground-level launches (initial height = 0) and elevated launches (initial height > 0). When launching from a height, the range increases compared to a ground-level launch with the same velocity and angle.