Projectile Motion Calculator — Range, Height & Time of Flight
This projectile motion calculator computes the horizontal range, maximum height, and time of flight from initial velocity, launch angle, and optional initial height. Enter a 45° angle to maximize range. Results assume ideal conditions with no air resistance and g = 9.80665 m/s².
Projectile Motion Calculator
Enter the initial velocity, launch angle, and optional height to calculate range, maximum height, and time of flight.
45° maximizes range
Frequently Asked Questions
What is projectile motion?
Projectile motion is the motion of an object thrown or projected into the air that moves under the sole influence of gravity (ignoring air resistance). The object follows a curved parabolic path. The horizontal velocity remains constant while the vertical velocity changes due to gravitational acceleration (9.80665 m/s²).
What angle maximizes the range of a projectile?
A launch angle of 45° maximizes the horizontal range of a projectile when launched from ground level with no air resistance. This is because 45° provides the optimal balance between horizontal velocity and time of flight. Complementary angles like 30° and 60° produce the same range but differ in maximum height and time aloft.
How do you calculate the range of a projectile?
Range = Vx × t, where Vx = v·cos(θ) is the horizontal velocity and t is the total time of flight. Time of flight is found by solving h + Vy·t − ½g·t² = 0, giving t = (Vy + √(Vy² + 2gh)) / g. For ground-level launch, this simplifies to t = 2Vy/g and Range = v²·sin(2θ)/g.
How do you calculate maximum height in projectile motion?
Maximum height is reached when the vertical velocity equals zero. The formula is: H_max = h + Vy² / (2g), where h is the initial height, Vy = v·sin(θ) is the initial vertical velocity, and g = 9.80665 m/s² is gravitational acceleration.
How do you calculate time of flight for a projectile?
Time of flight is calculated by solving the vertical position equation for when y = 0: h + Vy·t − ½g·t² = 0. Using the quadratic formula: t = (Vy + √(Vy² + 2gh)) / g. For launch from ground level (h=0), this simplifies to t = 2Vy/g = 2v·sin(θ)/g.
What are Vx and Vy in projectile motion?
Vx is the horizontal velocity component: Vx = v·cos(θ). It remains constant throughout the flight (no horizontal forces without air resistance). Vy is the initial vertical velocity component: Vy = v·sin(θ). It decreases linearly due to gravity at a rate of g = 9.80665 m/s² per second, reaching zero at max height then becoming negative on descent.
Does air resistance affect projectile motion?
Yes, significantly. Air resistance reduces range, shifts the optimal angle below 45° (typically to 30–40°), and creates an asymmetric trajectory where descent is steeper than ascent. This calculator uses the ideal model (no air drag) which is accurate for dense, slow-moving objects but less so for lightweight or high-speed projectiles like arrows or bullets.
How does initial height affect projectile range?
A higher launch height increases the total time of flight, which increases horizontal range. The time of flight formula t = (Vy + √(Vy² + 2gh)) / g shows that as h increases, t increases. Even launching horizontally from a height produces substantial range — for example, a projectile launched at 10 m/s from a 20 m cliff travels about 20 m horizontally.
How to Calculate Projectile Motion
Projectile Motion Equations
Projectile motion is governed by two independent components: horizontal (constant velocity) and vertical (constant acceleration due to gravity). With initial velocity v, launch angle θ, and initial height h:
Velocity Components
Vx = v · cos(θ)
Vy = v · sin(θ)
Maximum Height
Hmax = h + Vy² / (2g)
Time of Flight
t = (Vy + √(Vy² + 2gh)) / g
Derived by solving h + Vy·t − ½g·t² = 0
Horizontal Range
R = Vx · t
Where g = 9.80665 m/s² is the standard acceleration due to gravity.
Effect of Launch Angle
The launch angle has a profound effect on the trajectory shape and where the projectile lands.
| Angle | Range (v=20 m/s) | Max Height | Time of Flight |
|---|---|---|---|
| 15° | 20.36 m | 1.36 m | 1.06 s |
| 30° | 35.36 m | 5.10 m | 2.04 s |
| 45° | 40.77 m (max) | 10.19 m | 2.88 s |
| 60° | 35.36 m | 15.29 m | 3.53 s |
| 75° | 20.36 m | 19.03 m | 3.94 s |
Key Insight: 45° Maximizes Range
When launched from ground level with no air resistance, a 45° angle yields the maximum horizontal range. Complementary angles (e.g., 30° and 60°) produce the same range but differ in height and time of flight. Shallow angles travel farther along the ground; steep angles reach greater heights.
Air Resistance Note
This calculator uses the ideal projectile model — it assumes a vacuum with no air drag. In reality, air resistance significantly affects trajectories:
- Range is reduced — drag opposes motion, slowing the projectile and causing it to fall short of the ideal prediction.
- Optimal angle shifts below 45° — with air resistance, the range-maximizing angle is typically 30°–40° depending on the object.
- Asymmetric trajectory — the descent is steeper than the ascent because the projectile slows more during flight.
- Spin effects (Magnus effect) — spinning objects like balls curve due to pressure differences created by spin and airflow.
For engineering applications requiring precision (ballistics, aerospace, sports science), use computational models that incorporate drag coefficients, object shape, and atmospheric conditions.
Real-World Examples
Sports: Soccer Kick
A professional soccer player can kick a ball at roughly 30 m/s at a 30° angle. The ideal model predicts a range of about 79 m and a max height of 11.5 m — close to real-world free kicks before air drag and spin take over.
Engineering: Water Fountain Nozzle
A fountain jet at 5 m/s and 60° reaches a maximum height of about 0.95 m and lands 2.21 m away. Designers use these calculations to plan basin sizes and aesthetic arcs.
Physics Education: Cliff Drop
A ball rolled off a 20 m cliff at 10 m/s horizontally (angle approaches 0°, so enter a small angle like 1°) demonstrates how initial height dramatically extends range. Even a tiny vertical velocity component adds to flight time.
Military: Artillery Elevation
Artillery crews historically used 45° for maximum range under battle conditions. Modern systems account for air resistance, wind, and Earth's rotation (Coriolis effect), but the 45° baseline principle holds in ideal conditions.