Analyze parabolic projectile motion with precision. Calculate time of flight, maximum height, range, and final velocity components with step-by-step physics showing every calculation.
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⏱ Time of Flight
5.42 s
Total time in the air
📏 Maximum Height
34.5 m
Highest point of trajectory
🎯 Range
115.0 m
Horizontal distance traveled
💨 Final Velocity
30.0 m/s
Speed at impact
📐 Impact Angle
-45.0°
Below horizontal at impact
📝 Step-by-Step Calculation
Real-World Projectile Motion Examples
🔴 Cannonball Shot
A cannon fires a cannonball at 50 m/s at a 30° angle from ground level. Neglecting air resistance:
Parameter
Value
Initial Velocity
50.0 m/s
Launch Angle
30.0°
Initial Height
0.0 m
Vertical Component (v₀y)
50.0 × sin(30°) = 25.0 m/s
Horizontal Component (v₀x)
50.0 × cos(30°) = 43.3 m/s
Time of Flight
2 × 25.0 / 9.81 = 5.10 s
Maximum Height
25.0² / (2 × 9.81) = 31.9 m
Range
43.3 × 5.10 = 220.8 m
Final Velocity
50.0 m/s (same as initial due to symmetry)
Impact Angle
-30.0° (symmetrical to launch)
🏌️ Golf Ball Drive
A golfer drives a ball at 70 m/s at a 15° angle from a tee (ground level).
Horizontal component: 70.0 × cos(15°) = 67.6 m/s
Vertical component: 70.0 × sin(15°) = 18.1 m/s
Time of Flight: 2 × 18.1 / 9.81 = 3.69 s
Maximum Height: 18.1² / (2 × 9.81) = 16.7 m
Range: 67.6 × 3.69 = 249.4 m (≈273 yards — a solid drive!)
🏆 Result: The ball travels approximately 249 meters down the fairway.
🏀 Basketball Shot
A basketball player shoots from a height of 2.1 m at 8 m/s and 55° above horizontal. The hoop is at 3.05 m.
Parameter
Value
Initial Velocity
8.0 m/s
Launch Angle
55.0°
Initial Height
2.1 m
v₀x
8.0 × cos(55°) = 4.59 m/s
v₀y
8.0 × sin(55°) = 6.55 m/s
Time of Flight
(6.55 + √(6.55² + 2×9.81×2.1)) / 9.81 = 1.55 s
Maximum Height
2.1 + 6.55² / (2 × 9.81) = 4.29 m
Range
4.59 × 1.55 = 7.11 m
🏆 Result: The ball reaches 4.29 m (well above the 3.05 m hoop) and travels 7.11 m horizontally — a successful shot!
Projectile Motion Equations
Projectile motion describes the trajectory of an object launched into the air under the influence of gravity alone. The motion is analyzed by decomposing it into independent horizontal (constant velocity) and vertical (constant acceleration) components. Air resistance is neglected in the ideal model.
1. Velocity Components
The initial velocity is resolved into horizontal and vertical components:
v₀x = v₀ · cos(θ)
Horizontal component — constant throughout flight (no air resistance)
v₀y = v₀ · sin(θ)
Vertical component — changes due to gravity (a = -g)
2. Time of Flight
The total time the projectile stays in the air depends on initial height:
t = (v₀y + √(v₀y² + 2·g·h₀)) / g
When launched from height h₀ above ground (g = 9.81 m/s²)
t = 2 · v₀y / g
When launched from ground level (h₀ = 0) — symmetrical trajectory
3. Maximum Height
The highest point of the trajectory occurs when vertical velocity reaches zero:
h_max = h₀ + v₀y² / (2·g)
Maximum height reached above ground level
4. Range (Horizontal Distance)
The horizontal distance traveled is the horizontal velocity multiplied by time of flight:
R = v₀x · t
Range = horizontal velocity × time of flight
5. Final Velocity at Impact
The velocity at impact combines horizontal and vertical components:
v_final = √(v₀x² + v_y²)
Where v_y = v₀y − g·t
6. Impact Angle
θ_impact = arctan(v_y / v₀x)
Angle below horizontal (negative value indicates downward direction)
Key Principles
1
Independence of motion: Horizontal and vertical motions are analyzed separately — the horizontal velocity is constant, while vertical motion follows constant acceleration due to gravity.
2
Symmetry: For ground-level launches (h₀ = 0), the trajectory is symmetric — time up equals time down, and impact speed equals launch speed.
3
Optimal angle: For maximum range on flat ground, use a 45° launch angle. For elevated targets or initial heights, the optimal angle changes.
4
Gravity constant: All calculations use g = 9.81 m/s² (standard gravity on Earth's surface).
🚀
Full Trajectory Analysis
Compute time of flight, maximum height, range, final velocity, and impact angle with visual trajectory diagram.
📐
Step-by-Step Solution
Every calculation is broken down with formulas and intermediate values shown for complete understanding.
🔄
Multiple Unit Support
Works in m/s, km/h, ft/s, and mph for velocity. Height in meters or feet. Results adapt automatically.
📊
Visual Trajectory Plot
See the parabolic arc of your projectile drawn in real-time on an interactive diagram.
Projectile motion is the motion of an object thrown or projected into the air, subject only to the force of gravity. The object — called a projectile — follows a curved path known as a trajectory or parabolic arc. This fundamental concept in classical mechanics was first analyzed by Galileo Galilei in the 16th century and remains one of the most important topics in introductory physics.
The key insight in analyzing projectile motion is that the horizontal and vertical motions are independent of each other. The horizontal component of velocity remains constant (ignoring air resistance), while the vertical component is affected by gravity with a constant downward acceleration of g = 9.81 m/s² on Earth.
This principle of independence allows us to break any projectile problem into two simpler one-dimensional motion problems — one with constant velocity (horizontal) and one with constant acceleration (vertical).
Understanding Your Results
The Projectile Motion Calculator computes five key parameters based on your inputs:
Time of Flight — The total duration the projectile remains in the air, from launch to impact. For ground-level launches, this is simply twice the time to reach maximum height.
Maximum Height — The highest vertical position reached during the trajectory. This occurs when the vertical velocity component reaches zero at the apex of the parabola.
Range — The horizontal distance traveled from launch point to landing point. The range depends on both the horizontal velocity and the time of flight.
Final Velocity — The speed at which the projectile strikes the ground. For symmetrical trajectories (ground-to-ground), this equals the launch speed.
Impact Angle — The angle at which the projectile meets the ground, measured below the horizontal. For ground-level launches, this equals the negative of the launch angle.
Applications of Projectile Motion
Understanding projectile motion is essential in numerous real-world fields and activities:
🏅 Sports Science
From basketball shots and golf drives to javelin throws and soccer kicks, athletes and coaches use projectile motion principles to optimize performance.
🎯 Military & Defense
Ballistics calculations for artillery, missiles, and mortar fire rely on projectile motion equations to accurately predict impact points.
🚀 Aerospace Engineering
Rocket staging, re-entry trajectories, and payload delivery systems all begin with basic projectile motion analysis before adding air resistance and other factors.
🎢 Entertainment
Video game physics engines, animated film effects, and theme park ride designs all use projectile motion for realistic motion simulation.
Frequently Asked Questions
What is the optimal launch angle for maximum range?
For a projectile launched from and landing on flat ground (same elevation), the optimal launch angle for maximum horizontal range is 45 degrees. This occurs because sin(2θ) is maximized at θ = 45° in the range equation R = (v₀² × sin(2θ)) / g. However, if the launch and landing points are at different elevations (e.g., throwing from a cliff), the optimal angle changes.
Does air resistance affect projectile motion?
Yes, significantly. This calculator uses the ideal projectile motion model that neglects air resistance. In reality, air resistance (drag) reduces both the range and maximum height of a projectile. The effect is more pronounced for lighter objects, higher velocities, and less dense projectiles. For everyday objects thrown at moderate speeds (like a baseball), the ideal model provides a reasonable approximation — within about 10–20% of reality for range.
What does "initial height" mean and when should I use it?
Initial height (h₀) is the vertical distance from the launch point to the ground level. Use this field when the projectile is launched from an elevated position, such as a cannon on a hilltop, a basketball player shooting from above the floor, or a ball thrown from a balcony. If the projectile is launched from ground level (like a golf ball from a tee), set this value to 0.
Why are my results different at very high launch angles?
At very high angles (near 90°), the projectile goes nearly straight up and comes nearly straight back down, resulting in very small range but large maximum height. The time of flight is still calculated correctly. A launch angle of exactly 90° produces vertical motion only — the range will be zero and the projectile will land at the same spot it was launched.
What is the value of g used in calculations?
This calculator uses the standard acceleration due to gravity g = 9.81 m/s², which is the average value at Earth's surface. The actual value varies slightly depending on latitude (9.78 m/s² at the equator to 9.83 m/s² at the poles) and altitude. For most practical purposes, 9.81 m/s² provides accurate results.
Can this calculator be used for space or other planets?
The calculations use Earth's gravity (g = 9.81 m/s²). However, you can approximate projectile motion on other celestial bodies by considering the relative gravity. On the Moon (g ≈ 1.62 m/s²), a projectile would go about 6 times higher and 6 times farther. On Mars (g ≈ 3.72 m/s²), about 2.6 times farther. For precise results on other planets, you would need to adjust the gravity constant.
⚠️ Important Note: This Projectile Motion Calculator uses the ideal physics model that neglects air resistance. Real-world results may differ due to drag, wind, spin, and other environmental factors. For precise engineering or safety-critical applications, use more comprehensive models that account for these variables.