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2D Projectile Motion Trajectory Calculator

Classical Newtonian projectile motion describes the parabolic flight path of an object launched into a uniform gravitational field without aerodynamic drag.

Initial muzzle or launch velocity vector magnitude.

Elevation angle relative to the horizontal plane (0° to 90°).

Elevation of the launch platform above final ground impact level.

Local gravity (Earth standard 9.81 m/s², Moon 1.62 m/s², Mars 3.72 m/s²).

Calculated Result
40.77

Horizontal Range (m)

Maximum Apex Height

10.19 m

Total Flight Time

2.88 s

Impact Speed

20.00 m/s

Impact Angle

45.0°

Calculation Breakdown

  1. Initial Velocity Vector Componentsv_{0x} = 14.14 m/s, v_{0y} = 14.14 m/sv0x=v0cos⁡(θ),v0y=v0sin⁡(θ)v_{0x} = v_0 \cos(\theta), \quad v_{0y} = v_0 \sin(\theta)
  2. Time to Peak Elevation1.44 secondstpeak=v0ygt_{\text{peak}} = \frac{v_{0y}}{g}
  3. Maximum Trajectory Apex Height10.19 m (33.4 ft)Hmax=h0+v0y22gH_{\text{max}} = h_0 + \frac{v_{0y}^2}{2g}
  4. Total Flight Time to Ground Impact2.88 secondstf=v0y+v0y2+2gh0gt_f = \frac{v_{0y} + \sqrt{v_{0y}^2 + 2 g h_0}}{g}
  5. Horizontal Range40.77 m (133.8 ft)R=v0x×tfR = v_{0x} \times t_f
  6. Final Terminal Impact Velocity20.00 m/s @ 45.0° below horizontalvf=v0x2+vfy2v_f = \sqrt{v_{0x}^2 + v_{fy}^2}

Trajectory Apex Height vs Launch Angle (at 20 m/s)

Interactive visualization based on your current inputs

Peak Height (m)
0.05.110152015°30°45°60°75°90°Launch Angle (°)Peak Height (m)

What Is the 2D Projectile Motion Trajectory Calculator?

Projectile motion is the classic form of two-dimensional kinematics where an airborne object is subjected exclusively to the downward acceleration of gravity.

From artillery ballistics and athletic throws to spacecraft re-entry arcs, parabolic mechanics form the mathematical basis of trajectory engineering.

How Does the 2D Projectile Motion Trajectory Calculator Work?

The initial launch velocity is split into orthogonal vectors: vx = v₀ cos(θ) along the horizontal and vy = v₀ sin(θ) vertically.

Horizontal velocity remains constant, while vertical velocity undergoes steady gravitational deceleration: vy(t) = v₀ sin(θ) - gt.

Integrating these equations yields parabolic spatial coordinates (x(t), y(t)) describing the complete trajectory arc.

2D Projectile Motion Trajectory Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

R=v0xtf=(v0cos⁡θ)[v0sin⁡θ+v02sin⁡2θ+2gh0g]R = v_{0x} t_f = (v_0 \cos\theta) \left[\frac{v_0 \sin\theta + \sqrt{v_0^2 \sin^2\theta + 2 g h_0}}{g}\right]

Vertical kinematics determines time of flight by solving for zero height, while horizontal velocity remains constant in the absence of air resistance.

How to Use the 2D Projectile Motion Trajectory Calculator

  1. Enter the initial muzzle velocity in meters per second and the launch angle in degrees.
  2. If launching from a cliff or raised platform, enter the initial elevation above ground.
  3. Adjust local gravity if computing trajectories on extraterrestrial bodies like the Moon or Mars.

Step-by-Step Example Calculation

Standard Ground Launch at 45°

Input Values:

velocity:20
angle:45
initialHeight:0
gravity:9.81
Worked Steps: A 20 m/s launch at 45° lands 40.77 meters away with an apex height of 10.19 m and flight time of 2.88 s.

Understanding Your Result

Horizontal Range: Total linear distance traversed over ground before impact.

Apex Height: The highest elevation attained above the initial launch elevation.

Flight Time: Total seconds elapsed between launch and ground impact.

Impact Velocity: Vector sum of horizontal and terminal vertical velocities upon touchdown.

Factors That Affect the Result

  • Launch Velocity: Range scales quadratically with initial speed (doubling velocity quadruples range).
  • Elevation Angle: Directs the balance between hang time and horizontal forward speed.
  • Gravity Magnitude: Weaker gravity extends flight time and dramatically increases distance.

When Should You Use This Calculator?

  • Physics education and introductory classical mechanics homework problems.
  • Engineering estimation of ballistic flight trajectories and sports ball mechanics.
  • Game development trajectory modeling and robotic catapult design.

Assumptions & Limitations

  • Zero aerodynamic drag, wind speed, or Magnus spin forces are accounted for.
  • Gravitational acceleration is assumed uniform and parallel (flat-Earth approximation).
  • Earth rotational Coriolis deflection is neglected.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Exact analytical closed-form solution to vacuum Newtonian kinematics with 100% mathematical precision.

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