2025-08-21 14 min
Projectile Motion Calculator
Analyze projectile trajectories: range, max height, time of flight, impact velocity. Supports elevated launches and variable gravity.
Calculator
Unit System
Preset
Initial Velocity(m/s) Help
m/s
Launch speed
Launch Angle(°) Help
°
0-90 degrees
Initial Height(m) Help
m
Launch elevation
Gravity(m/s²) Help
m/s²
Earth: 9.81, Moon: 1.62
Decimal Precision
Results update live as you type
Results
Normal
Range
254.84m
Max Height
63.71m
Time of Flight
7.21s
Impact Velocity
50m/s
Impact Angle
45°
v_x (horizontal)
35.36m/s
v_y₀ (vertical)
35.36m/s
Trajectory Diagram
Formula
Range, max height, and time of flight for projectile motion on level ground.
R = v² × sin(2θ) / g
Alternative Form
H = v² × sin²(θ) / (2g), T = 2v × sin(θ) / g
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| R | Range | m | Horizontal distance traveled |
| H | Max Height | m | Maximum vertical height |
| T | Time of Flight | s | Total time in air |
| v | Initial Velocity | m/s | Launch speed |
| θ | Launch Angle | ° | Angle from horizontal |
| g | Gravity | m/s² | Gravitational acceleration |
Explanation
Range is maximized at 45°. Height scales with sin²(θ). Time scales with sin(θ).
Derivation
Decompose v into vx = vcosθ and vy = vsinθ. Horizontal: x = vxt. Vertical: y = vy*t - ½gt².
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Step 1
Enter launch velocity and angle
Result: Parameters entered
Step 2
Set initial height and gravity
Result: Conditions set
Step 3
Compute trajectory equations
Result: Calculations performed
Step 4
Read range, height, time, and impact data
Result: Results displayed
Final Answer
254.84
Worked Examples
See how the calculator works with different scenarios.
Scenario
50 m/s at 45° from ground level.
Inputs
Velocity:50 m/s
Angle:45°
Height:0 m
Gravity:9.81 m/s²
Calculation
R = 50² × sin(90°) / 9.81 = 255 m
Result
Range = 255 m
Interpretation
Optimal angle gives maximum range.
Understanding Your Result
Your Result: Range: 254.84 m
Typical range: Basketball: 4-5 m. Golf: 250-300 m. Artillery: 20-40 km. ICBM: 10,000+ km.
What This Means
The trajectory is a parabola determined by initial velocity, angle, and gravity.
Why It Matters
Essential for sports, ballistics, aerospace, and fountain design.
Professional Interpretation
Compare trajectory against target requirements. Consider drag for precision.
Recommendations
- •45° maximizes range on level ground
- •Consider air resistance for light/fast objects
- •Adjust for elevated launch points
- •Use variable gravity for other planets
Educational Content
Introduction
Projectile motion describes objects launched under gravity alone, following parabolic paths.
Concept Explanation
Horizontal motion is constant velocity. Vertical motion is constant acceleration (g). The path is a parabola.
Applications
▸Sports analysis
▸Military ballistics
▸Aerospace trajectory
▸Fountain design
▸Volcanic ejecta
Benefits
- •Predictable trajectories
- •Fundamental physics model
- •Applicable across scales
- •Foundation for advanced ballistics
Limitations
- •Ignores air resistance
- •Assumes flat Earth
- •No wind effects
- •Point mass only
Best Practices
- •45° for max range on level ground
- •Account for drag at high speeds
- •Use elevated launch formulas for real scenarios
- •Consider Coriolis for long range
Common Misconceptions
Myth: 45° always gives max range
Truth: Only on level ground. For elevated launches, the optimal angle decreases.
Practical Tips
- •45° maximizes range on level ground
- •Heavier objects are less affected by drag
- •Moon gravity = 1.62 m/s² (6x range)
- •Complementary angles give same range
Common Mistakes
Ignoring air resistance
Add drag model for light/fast projectiles.
Using 45° for elevated launch
Optimal angle is lower for elevated launches.
Assumptions
This calculator makes the following assumptions to provide estimates:
No Air Resistance: Vacuum trajectory assumed.
Point Mass: Object treated as a particle.
Constant Gravity: g is constant throughout flight.
Limitations
Be aware of these limitations when using this calculator:
Idealized: Real trajectories affected by drag, wind, spin.
Flat Earth: Curvature matters for ranges > 10 km.
Frequently Asked Questions
Answers to common questions about this calculator.
What is projectile motion?
Projectile motion describes the motion of an object launched into the air under only the influence of gravity (ignoring air resistance). The path followed is a parabola. The motion can be decomposed into independent horizontal (constant velocity) and vertical (constant acceleration) components that occur simultaneously.
What is the optimal launch angle for maximum range?
For a projectile launched from ground level, the optimal angle for maximum horizontal range is 45° (ignoring air resistance). This angle equally divides the initial velocity between horizontal and vertical components. For elevated launches, the optimal angle decreases slightly below 45°.
What is the range formula for projectile motion?
The horizontal range is given by R = (v² × sin(2θ)) / g, where v is the initial velocity, θ is the launch angle, and g is gravitational acceleration. This formula assumes level ground and no air resistance. For elevated launches, the quadratic equation must be solved for time of flight first.
What is the maximum height formula?
The maximum vertical height reached is H = (v² × sin²(θ)) / (2g) for level-ground launches. For elevated launches from height h₀, the maximum height is H = h₀ + (v² × sin²(θ)) / (2g). The maximum height occurs at the trajectory apex where vertical velocity becomes zero.
What is the time of flight?
For level-ground launches, time of flight is T = (2v × sin(θ)) / g. For elevated launches, T = (v·sin(θ) + √(v²·sin²(θ) + 2g·h₀)) / g, solving the quadratic equation 0 = h₀ + v·sin(θ)·t - ½g·t².
How does launch angle affect trajectory shape?
Low angles (near 0°) produce flat, shallow trajectories with minimal height and moderate range. The 45° angle maximizes range. High angles (above 45°) produce steep, arcing trajectories with greater height but shorter range. Complementary angles (θ and 90°-θ) produce the same range (sin(2θ) = sin(2(90°-θ))).
How does initial velocity affect projectile motion?
Range scales with v² (doubling velocity quadruples the range). Maximum height also scales with v². Time of flight scales linearly with v. These relationships make initial velocity the most influential parameter in projectile motion.
How would projectile motion differ on the Moon?
The Moon's gravity is about 1/6 of Earth's (1.62 vs 9.81 m/s²). For the same launch conditions, a projectile on the Moon travels about 6 times farther horizontally, reaches about 6 times higher, and stays in the air about 6 times longer compared to Earth.
What is the trajectory equation (parabolic path)?
The trajectory equation is y = x·tan(θ) - (g·x²) / (2v²·cos²(θ)), where y is vertical position and x is horizontal position. This is a quadratic function of x (parabola). The coefficient a = -g/(2v²cos²θ) determines the downward curvature of the path.
How does air resistance affect projectile motion?
Air resistance (drag) reduces range, maximum height, and time of flight compared to the ideal vacuum case. Drag force is approximately proportional to v² and acts opposite to velocity, causing asymmetric trajectories with steeper descents than ascents. Heavy, dense objects (cannonballs) are less affected than light objects (feathers).
What is the horizontal component of velocity?
The horizontal velocity component is v_x = v·cos(θ) and remains constant throughout flight (in vacuum). This is because no horizontal forces act on the projectile in the ideal case. The horizontal displacement at any time is simply x = v·cos(θ)·t.
What is the vertical component of velocity?
The vertical velocity component is v_y = v·sin(θ) - g·t. It decreases linearly with time due to gravity. At the apex, v_y = 0. On descent, v_y becomes negative (downward). The vertical position is y = h₀ + v·sin(θ)·t - ½g·t².
What is impact velocity?
Impact velocity is the speed at which the projectile strikes the ground. For level launches (same height), the impact speed equals the launch speed due to energy conservation, but the vertical component is reversed. For elevated launches, impact speed is higher: v_impact = √(v²·cos²(θ) + (v·sin(θ))² + 2g·h₀).
What is impact angle?
The impact angle is the angle below horizontal at which the projectile strikes the ground. For level launches, it equals the launch angle. For elevated launches, the impact angle is steeper than the launch angle because the projectile gains additional downward velocity from the fall height.
What is the range of a basketball free throw?
A free throw releases from about 2.1 m height, at about 52° angle, with about 8.5 m/s velocity, traveling 4.57 m horizontally to the hoop center. The ball enters the basket at a downward angle of about 45-50°, passing through the hoop with minimal clearance.
What is the range of a soccer goal kick?
Professional soccer goal kicks achieve ranges of 70-90 m (77-98 yards). An initial velocity of 28 m/s (100 km/h) at 30-35° launch angle produces a trajectory reaching about 10 m in height with total time of flight around 3 seconds.
What is the range of a golf drive on the PGA Tour?
PGA Tour professionals drive the ball 290-320 yards (265-293 m) on average. Initial ball speed is 70-80 m/s (252-288 km/h), launch angle 10-14°, and backspin 2,500-3,500 rpm. The dimpled surface and spin create lift that increases carry distance by about 30-40% compared to a smooth sphere.
What is the range of a howitzer artillery shell?
Modern 155 mm howitzers like the M777 fire shells to ranges of 30-40 km (18-25 miles) with rocket-assisted projectiles extending to 50+ km. Muzzle velocity is about 827 m/s (Mach 2.4) at a typical launch angle of 45-55° depending on the target relative location.
How does the Coriolis effect affect projectiles?
The Coriolis effect deflects projectiles to the right in the Northern Hemisphere and left in the Southern Hemisphere due to Earth's rotation. For a 20 km artillery shot at mid-latitudes, the deflection is about 50-100 m. Snipers and artillery crews must account for Coriolis at ranges over 1 km.
What is the difference between ballistic and guided projectiles?
Ballistic projectiles follow a purely ballistic trajectory determined only by initial conditions, gravity, and drag. Guided projectiles have fins, control surfaces, or thrusters that actively alter the trajectory in flight. Modern smart artillery shells can adjust trajectory to hit within 10-20 m of target compared to 200-300 m for unguided shells.
How does altitude affect projectile range?
Higher altitudes have lower air density, reducing drag significantly. A projectile fired at 3,000 m altitude might achieve 15-20% greater range than the same shot at sea level due to reduced aerodynamic drag. This is a significant consideration for military operations in mountainous regions.
What is the maximum range of a thrown object by a human?
Javelin world record is 98.48 m (322 ft, 10 in). Discus (2 kg) record is 74.08 m (243 ft). Hammer (7.26 kg) record is 86.74 m (284 ft). Shot put (7.26 kg) record is 23.37 m (76.7 ft). Cricket ball throw record is about 140 m but this is a weighted implement.
What is the optimal angle for maximum height?
The optimal launch angle for maximum height is 90° (straight up). The maximum height formula H = v²sin²(θ)/(2g) is maximized when sin(θ) = 1, i.e., θ = 90°. However, launching straight up gives zero range, making it useless for distance applications.
How does mass affect projectile motion in the ideal case?
In the ideal vacuum case, mass does NOT affect projectile motion. All objects, regardless of mass, follow the same parabolic trajectory when launched with identical velocity and angle. This was famously demonstrated by Apollo 15 astronaut David Scott who dropped a hammer and feather on the Moon.
What is the range of an ICBM?
Intercontinental Ballistic Missiles (ICBMs) have ranges of 5,500-15,000+ km. They follow a highly elliptical trajectory (suborbital), reaching apogee altitudes of about 1,200-1,500 km. Re-entry vehicles impact at speeds of about 7 km/s (Mach 20+). The flight time from launch to impact is about 30-35 minutes.
What is the projectile motion of a rocket?
Rockets differ from simple projectiles because they have continuous thrust, changing mass, and often guidance. However, after engine cutoff (burnout), the rocket follows a ballistic trajectory described by projectile motion equations. The coasting phase between burns follows the basic projectile model.
How does spin affect a projectile?
Spin stabilizes a projectile through gyroscopic effects (like a football or bullet). Spin also generates Magnus force (lift or curve) depending on spin direction relative to velocity. Backspin on a golf ball generates lift, increasing range. Topspin on a tennis ball causes downward force, making the ball drop faster.
What is the equation for projectile motion with drag?
With quadratic drag (F_drag = ½ρCdAv²), the equations become coupled nonlinear ODEs with no closed-form solution: dv_x/dt = -kv·v_x and dv_y/dt = -kv·v_y - g, where k = ½ρCdA/m. Numerical integration (e.g., Runge-Kutta methods) is required for accurate solutions.
How do you calculate projectile motion on uneven terrain?
For uneven terrain with landing height different from launch, solve the quadratic ½gt² - v·sin(θ)·t - (h_launch - h_land) = 0 for t, then use the positive root. Compute range as v·cos(θ)·t. The impact angle becomes arctan(|v_y|/v_x) where v_y = v·sin(θ) - g·t.
What is the muzzle velocity of common firearms?
Typical muzzle velocities: .22 LR: 340 m/s, 9mm Parabellum: 360 m/s, .45 ACP: 255 m/s, 5.56×45mm NATO (M16): 950 m/s, 7.62×51mm NATO (.308): 860 m/s, .50 BMG: 880 m/s, 12-gauge shotgun slug: 500 m/s. These velocities are at the muzzle and decrease downrange.
What is the trajectory of a baseball pitch?
A fastball thrown at 95 mph (42.5 m/s) from 60.5 ft (18.44 m) reaches home plate in about 0.4 seconds. The ball drops about 2-3 m due to gravity during flight. The trajectory actually rises slightly initially because the release point is above the mound and home plate is at ground level.
How does the Magnus effect affect projectile motion?
The Magnus effect causes a spinning projectile to curve in the direction of the spin. For a ball with topspin, the pressure difference creates a downward force, making it drop faster. For backspin, lift is generated (golf ball). Sidespin causes left/right curve (soccer, baseball curveball).
What is the optimal launch angle for minimum time of flight?
The minimum time of flight for a given range occurs at a 45° launch angle. For a given initial velocity, the time of flight is T = 2v·sin(θ)/g, which increases with launch angle (zero range at 0°, maximum time at 90°). For hitting a target at known distance, there are typically two possible angles.
What is the range of a thrown baseball from the outfield?
Major League outfielders can throw a baseball about 90-120 m (300-400 ft) on the fly. The longest recorded throws exceed 130 m (430 ft). Outfield velocity is typically 35-40 m/s (78-90 mph) with a launch angle of about 30-35° to maximize distance.
How does projectile motion apply to water fountains?
Water fountains are practical applications of projectile motion. Water exits nozzles at velocities of 5-20 m/s at angles typically 45-75°. The height and spread pattern of a fountain are determined by the same projectile equations. Higher angles create taller, narrower patterns; lower angles create shorter, wider patterns.
What is the range of a mortar shell?
Mortars are high-angle artillery weapons firing at angles typically 45-85°. A 60 mm mortar has range 1,500-3,500 m. An 81 mm mortar reaches 3,000-6,000 m. A 120 mm mortar can fire 5,000-8,000 m. The high launch angle enables the projectile to clear obstacles and strike behind cover.
What is the velocity of a projectile at the top of its trajectory?
At the apex (maximum height), the vertical velocity component is zero (v_y = 0). Only the horizontal component remains: v = v·cos(θ). This is the minimum speed during the trajectory (in vacuum). The remaining kinetic energy at the apex is KE = ½m(v·cos(θ))².
What is the optimal launch angle for a basketball free throw?
The optimal release angle for a basketball free throw is about 52-55° above horizontal, with the release point at about 2.1 m height and hoop at 3.05 m. The ball should enter the hoop at an angle of about 45-50° to maximize the effective opening size. The launch speed required is about 7.5-8.5 m/s depending on the exact angle and release height. A higher arc provides a larger margin of error but requires more precise speed control.
What is the range of a trebuchet or medieval siege weapon?
Medieval trebuchets could launch projectiles weighing 50-150 kg to ranges of 200-400 m. The largest trebuchets at sieges like Kenilworth (1266 CE) could throw 100-200 kg stones approximately 300-400 m. The projectile motion followed a high-arcing trajectory (launch angle about 45-55°), with the counterweight (10-20 times the projectile mass) providing the necessary kinetic energy through gravitational potential energy conversion.
How does the Earth's curvature affect long-range projectile motion?
For ranges exceeding about 10 km, the Earth's curvature becomes significant. The ground "falls away" from the projectile at a rate of about 0.08 m per km of horizontal distance (parabolic approximation). Artillery fire control systems must account for this: the apparent range is longer than the geometric range because the target is below the local horizontal plane at the firing position. For ICBM ranges of 5,000-15,000 km, the Earth's curvature dominates the trajectory analysis, which becomes an orbital mechanics problem.
What is the launch angle of a javelin?
Elite javelin throwers release the javelin at an angle of about 34-38° above horizontal, with the javelin itself angled slightly above the flight path for aerodynamic efficiency. The release velocity is about 28-32 m/s (100-115 km/h) for world-class athletes. The aerodynamic design of the javelin (which was modified in 1986 to shorten range for safety) creates lift that extends the range beyond what a purely ballistic trajectory would predict.
What is the projectile motion of a water droplet from a sprinkler?
Impact sprinklers throw water droplets in a pattern determined by projectile motion combined with wind effects and droplet evaporation. The droplets leave the sprinkler nozzle at velocities of 5-15 m/s at angles typically between 20-35° for maximum coverage. Drop size distribution affects the range: larger droplets (2-5 mm diameter) travel farther due to higher momentum and lower drag-to-mass ratio, while smaller droplets (0.5-1 mm) are more affected by wind and evaporation.
What is the maximum range for a given initial velocity?
The maximum range on level ground for a given initial velocity v is R_max = v²/g, achieved at θ = 45°. For example: v = 10 m/s gives R_max = 10.2 m, v = 50 m/s gives 255 m, v = 100 m/s gives 1,020 m, v = 200 m/s gives 4,080 m, and v = 1,000 m/s gives about 102 km. This formula assumes no air resistance and landing at the same height as launch. The quadratic dependence on velocity means that doubling the launch speed quadruples the maximum range.
What is the range of a potato gun or spud cannon?
A typical pneumatic potato gun launches a potato at 30-60 m/s (108-216 km/h) depending on pressure and barrel length. At a 45° launch angle, the theoretical range in vacuum would be 92-367 m. Actual ranges are significantly shorter (50-150 m) due to the high drag on the irregularly shaped, low-density projectile. The potato experiences high deceleration due to drag, with the velocity decreasing rapidly after launch.
How does wind affect projectile motion?
Wind adds a horizontal velocity component to the projectile relative to the ground. A crosswind causes lateral deflection (drift), while headwind reduces range and tailwind increases it. The effect is approximately proportional to wind speed and time of flight. For a typical soccer kick with 3-second flight time in a 20 km/h crosswind, the lateral deflection is about 15-20 m. Snipers and long-range shooters must account for wind using windage adjustments, with effects becoming significant beyond 300-500 m.
What is the optimum launch angle for water jet cutting?
In abrasive water jet cutting, the nozzle typically angles at 2-10° from vertical to optimize cutting efficiency and minimize kerf width. The water velocity at the nozzle can reach 800-1,000 m/s (Mach 2.5-3), driven by pressures of 3,800-6,200 bar. The trajectory of the abrasive particles within the jet follows a complex multiphase flow path rather than simple projectile motion, with particle acceleration and fragmentation occurring within the cutting stream.
What is the range of a slingshot or catapult?
Modern high-performance slingshots can launch projectiles at velocities of 50-70 m/s (180-250 km/h), achieving ranges of 100-200 m at optimal 45° launch angle. The velocity depends on the elastic properties of the bands (latex or surgical tubing), the draw length, and the projectile mass (typically 8-12 mm steel balls). The trajectory is significantly affected by drag due to the relatively low ballistic coefficient of small spherical projectiles.
What is the projectile motion of a rocket firework?
Firework rockets follow a two-phase trajectory: first, the rocket motor provides thrust during ascent, then after burnout the payload follows a ballistic trajectory to its burst point. Display shells are typically launched at 70-85° angles to achieve heights of 100-500 m before bursting. The initial velocity from the mortar tube is 20-50 m/s depending on the shell size and propellant charge. The burst point timing is set by a fuse calibrated to the expected time of flight.
How does temperature affect air density and projectile range?
Warmer air is less dense than cold air, reducing drag on projectiles. On a 35°C day, the range of a golf drive can be 5-10 m longer than on a 10°C day due to lower air density (about 5-8% less drag). Conversely, high humidity slightly reduces air density (water vapor is lighter than dry air), marginally increasing range. Air pressure also affects range: at higher altitudes (lower pressure), range increases significantly. For golf at 1,500 m elevation, drives are about 10% longer than at sea level.
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Glossary
Technical terms used in mortgage calculations explained in plain language.
- Range
- Horizontal distance from launch to impact.
- Trajectory
- The parabolic path of a projectile.
- Launch Angle
- Angle of initial velocity above horizontal.
- Ballistic Coefficient
- Measure of ability to overcome air resistance.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
ISO 80000-3
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Page Information
Last Updated:
2025-08-21
Version:
2.0.0
References:
License:
Free to use for educational and professional purposes.