Physics Calculator
Projectile Motion Calculator
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Solve 2D kinematics and projectile motion problems effortlessly. Enter the initial velocity, launch angle, and launch height to calculate the maximum horizontal range, maximum altitude, time of flight, and visualize the trajectory using our interactive chart.
Projectile Motion Calculator
Calculate trajectory, range, and flight time
Results
Max Range
40.77m
Flight Time
2.88s
Max Height
10.19m
Time to Peak
1.44s
Anatomy of a Projectile's Flight
The launch velocity splits into a horizontal component that stays constant and a vertical component that gravity slows, stops at the peak, and reverses. Together they trace a parabola.
What is Projectile Motion?
Projectile motion is a foundational concept in classical mechanics describing how an object—referred to as a projectile—moves along a curved, two-dimensional path called a trajectory under the sole influence of gravity. The study of this motion is crucial in fields ranging from ballistics and sports science to aerospace engineering and planetary physics.
In ideal projectile motion, we analyze the path of the object by breaking it down into two completely independent components:
- Horizontal Motion (X-axis): Since we neglect air resistance (aerodynamic drag), no horizontal force acts on the object. Consequently, the horizontal velocity remains perfectly constant throughout the flight ($a_x = 0$).
- Vertical Motion (Y-axis): The object experiences a constant downward force due to gravity. This causes a constant downward acceleration ($a_y = -g$), meaning the vertical velocity changes linearly over time, slowing down as it ascends and speeding up as it falls back to earth.
By treating these two axes independently, we can model the entire parabolic flight path using standard kinematic formulas.
Kinematic Equations & Trajectory Math
This calculator uses standard kinematic equations of motion. First, we resolve the initial velocity vector v₀ into its horizontal (v₀ₓ) and vertical (v₀y) components using the launch angle (θ):
Using these components, the calculator determines the following critical flight characteristics:
Launch Angle vs. Horizontal Range (v₀ = 20 m/s, h₀ = 0 m)
Review the theoretical range results achieved on Earth at different launch angles when using an initial speed of 20 m/s from ground level:
| Launch Angle (θ) | Max Range (m) | Max Height (m) | Flight Time (s) |
|---|---|---|---|
| 15° | 20.39 m | 1.37 m | 1.06 s |
| 30° | 35.31 m | 5.10 m | 2.04 s |
| 45° (Optimal Angle) | 40.77 m | 10.19 m | 2.88 s |
| 60° | 35.31 m | 15.29 m | 3.53 s |
| 75° | 20.39 m | 19.02 m | 3.94 s |
Gravitational Forces on Different Celestial Bodies
Because gravity ($g$) directly decelerates vertical motion, the celestial body on which a projectile is launched changes its flight trajectory drastically:
| Celestial Body | Gravity (m/s²) | Relative Strength (vs. Earth) |
|---|---|---|
| Moon | 1.62 m/s² | 0.165x |
| Mars | 3.71 m/s² | 0.378x |
| Earth | 9.81 m/s² | 1.000x |
| Jupiter | 24.79 m/s² | 2.527x |
Key Variables
- Initial Velocity (v₀): The speed at which the projectile is launched. Measured in meters per second (m/s).
- Launch Angle (θ): The angle relative to the horizontal. Measured in degrees. 45° yields the maximum range on flat ground.
- Initial Height (h₀): The starting elevation of the projectile. Measured in meters (m).
- Gravity (g): The acceleration due to gravity. Standard Earth gravity is 9.80665 m/s².
Example Calculations
Verify the mathematical breakdowns for standard projectile motion scenarios.
Example Scenario 1 — Soccer Ball Kick (Flat Ground)
Initial Velocity = 15 m/s, Launch Angle = 30°, Initial Height = 0 m, Gravity = 9.81 m/s²
Horizontal Velocity (v₀ₓ): 15 × cos(30°) = 12.99 m/s
Vertical Velocity (v₀y): 15 × sin(30°) = 7.50 m/s
Time to Peak (t_peak): 7.50 / 9.81 = 0.76 seconds
Total Flight Time (t_flight): 2 × 0.76 = 1.53 seconds
Maximum Altitude (H): 0 + (7.50²) / (2 × 9.81) = 2.87 meters
Maximum Range (R): 12.99 × 1.53 = 19.86 meters
Example Scenario 2 — Projectile Launched From Cliff
Initial Velocity = 10 m/s, Launch Angle = 45°, Initial Height = 20 m, Gravity = 9.81 m/s²
Horizontal Velocity (v₀ₓ): 10 × cos(45°) = 7.07 m/s
Vertical Velocity (v₀y): 10 × sin(45°) = 7.07 m/s
Time to Peak (t_peak): 7.07 / 9.81 = 0.72 seconds
Maximum Altitude (H): 20 + (7.07²) / (2 × 9.81) = 22.55 meters
Quadratic equation to solve: -4.905t² + 7.07t + 20 = 0
Total Flight Time (t_flight): ~2.86 seconds
Maximum Range (R): 7.07 × 2.86 = 20.22 meters
Frequently Asked Questions
- What is projectile motion?
- Projectile motion is a form of motion experienced by an object or particle (a projectile) that is launched near the Earth's surface and moves along a curved path under the sole influence of gravity (assuming air resistance is negligible).
- What assumptions does this calculator make?
- This calculator models ideal projectile motion: zero air resistance or aerodynamic drag, a constant acceleration due to gravity (9.81 m/s² by default), a flat and non-rotating Earth, and a point-mass representation of the projectile.
- What is the optimal angle for maximum range?
- When launched and landed on flat ground (initial height = 0), the maximum range is always achieved at exactly 45 degrees. If launched from an elevated position (height > 0), the optimal launch angle is slightly less than 45 degrees to allow horizontal velocity to act longer.
- How does initial height affect launch range and angle?
- An elevated starting position increases the time of flight, which allows the projectile to travel further horizontally. Consequently, the optimal angle for maximum range drops below 45 degrees as initial elevation increases.
- Why is the trajectory of a projectile parabolic?
- The trajectory is parabolic because horizontal motion proceeds at a constant velocity (zero acceleration), whereas vertical motion is subject to a constant gravitational acceleration. Combining these parametric functions of time (x = vt, y = vt - 0.5gt²) yields a quadratic equation of y in terms of x.
- How does gravity affect projectile motion on different planets?
- Higher gravity (like Jupiter's 24.79 m/s²) pulls the projectile down rapidly, reducing both height and range. Lower gravity (like the Moon's 1.62 m/s²) allows the projectile to stay aloft much longer, resulting in much higher heights and longer ranges.
- What is the velocity of the projectile at its highest point?
- At the peak of its trajectory, the vertical component of velocity (vy) is exactly 0. However, the projectile's total velocity is not zero; it is equal to its constant horizontal velocity component (vx = v0 * cos(theta)).
- Does the mass of the projectile affect its flight path?
- In an ideal physics vacuum (no air resistance), mass has no effect on the trajectory, range, height, or flight time. All objects fall with the same gravitational acceleration regardless of mass, as proven by Galileo.
- How does air resistance change the real-world trajectory?
- Air resistance (drag) opposes the motion, decreasing both horizontal and vertical velocity. This shortens the range and peak height, making the flight path asymmetrical and steeper during the descent phase.
- What are the horizontal and vertical acceleration components of a projectile?
- For an ideal projectile, horizontal acceleration is exactly zero (ax = 0 m/s²) because no horizontal forces act on it. Vertical acceleration is constant and directed downwards, equal to gravity (ay = -g, or -9.81 m/s² on Earth).