Range
R = v₀² sin(2θ) / gLast updated:
Projectile Motion Calculator
Calculate projectile range, maximum height, and flight time. Analyze ballistic trajectories with launch angle and initial velocity. Free physics kinematics t...
Projectile Motion Calculator
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Calculate range, max height, and flight time for projectiles
Earth: 9.81 | Moon: 1.62 | Mars: 3.72
Note: This calculator assumes no air resistance. For maximum range on level ground, use a 45° launch angle.
Enter values above to see results.
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Classical Kinematics
Projectile Motion Calculator: Range, Height & Trajectory Guide
Calculate range, maximum height, flight time, and trajectory for any launch angle and initial speed in ideal (drag-free) conditions.
Max Range Angle
45° (level ground)
Range
v₀² sin(2θ)/g
Max Height
v₀² sin²(θ)/2g
Flight Time
2v₀ sin(θ)/g
Reviewed by: CalculatorApp Physics & Engineering Team
What Is Projectile Motion?
Projectile motion describes the curved path of an object launched with an initial velocity under the sole influence of gravity. The horizontal and vertical components are independent: horizontal velocity is constant (no air resistance), while vertical velocity changes at 9.81 m/s² downward. This model underlies ballistics, sports science, and space trajectory planning.
Max Height
H = v₀² sin²(θ) / 2gFlight Time
T = 2v₀ sin(θ) / gHoriz. Velocity
v_x = v₀ cos(θ)| Launch Angle | Relative Range | Best Use |
|---|---|---|
| 15° / 75° | 50% of max range | Low/high trajectory same range |
| 30° / 60° | 87% of max range | Sports throws, medium arcs |
| 45° | 100% max range | Optimal flat-ground distance |
| 90° | 0 range (straight up) | Maximum height, zero range |
History Timeline
~330 BC: Aristotle describes projectile motion incorrectly as needing continuous force.
1638: Galileo proves parabolic trajectory and component independence in Two New Sciences.
1687: Newton's laws formalize gravitational acceleration as the sole vertical force.
1700s-1800s: Ballistic tables developed for artillery, applying projectile motion to warfare.
1900s: Wind tunnels and drag studies extend beyond ideal projectile model for aviation.
Modern era: Computer trajectory simulation and GPS-guided munitions build on classical equations.
NASA Trajectory Tools
Orbital and atmospheric trajectory resources and research.
PubMed Sports Science
Biomechanics and sports projectile motion research database.
NIST Physical Constants
Standard gravity and fundamental physical constants.
CDC Injury Ballistics
Ballistic injury epidemiology and prevention research.
Myth: 45 degrees always gives maximum range.
Fact: 45° is optimal only for level launch and landing. Different heights require a different optimal angle.
Myth: Air resistance is always negligible.
Fact: At high speeds or for lightweight objects, air drag significantly reduces range compared to ideal calculations.
Myth: Horizontal and vertical velocities are linked during flight.
Fact: They are completely independent; horizontal is constant, vertical changes at 9.81 m/s².
Myth: Two objects thrown at different angles but same speed land at different times always.
Fact: Complementary angles (e.g., 30° and 60°) give equal range but different flight times and heights.
FAQ (12)
What is projectile motion?
Projectile motion is 2D kinematics where a launched object follows a parabolic path under gravity with no air resistance.
What launch angle gives maximum range?
45 degrees gives maximum range on level ground when launch and landing heights are equal.
How do I find maximum height?
H = v₀² sin²(θ) / (2g). Height depends on the vertical component of initial velocity.
How is range calculated?
R = v₀² sin(2θ) / g. For level ground the range is symmetric around the 45-degree optimum.
What is total flight time?
T = 2v₀ sin(θ) / g. It is twice the time to reach maximum height.
Does horizontal velocity change during flight?
No. Without air resistance, horizontal velocity v_x = v₀ cos(θ) remains constant throughout the flight.
How does air resistance affect results?
Air resistance reduces range and height, especially at high speeds. This calculator assumes vacuum (no drag).
What is the velocity at any point?
v = √(v_x² + v_y²) where v_y = v₀ sinθ − gt. The horizontal component stays constant.
Can this apply to sports?
Yes. Baseball, basketball, soccer, and javelin throwing all involve projectile motion approximated by this model.
What is the complementary angle rule?
Two launch angles that add to 90 degrees produce the same range; e.g., 30° and 60° give equal range.
How does gravity change on other planets?
Replace g with the local gravitational acceleration (e.g., Mars: 3.72 m/s²) to model off-Earth trajectories.
What units should I use?
Use consistent SI units: meters per second (m/s) for velocity and degrees or radians for angle.
References
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