Projectile Motion & Parabolic Motion Calculator: Solving 2D Kinematics Problems for Physics

Kinematics, the study of motion, is one of the more grueling units in any physics course.  Often, the curriculum starts out fairly straight forward distinguishing between:

Definition Examples
Scalar quantities Measurements/values that are never negative
  • Driving 5 miles
  • Distance
  • Speed
Vector quantities Measurements/values that include direction [usually indicated by a positive or negative (+/-) sign]
  • Driving 5 miles North
  • Position
  • Velocity

From there, most courses define functions and equations for position, velocity, and acceleration, eventually concluding with the four essential 1-dimensional kinematic formulas:

vf = v0 + at vf2 = v02 + 2ad
d = v0t + (1/2)at2 d = (1/2)(vf + v0) t

Note: Some books/teachers use the following conventions instead:

  • v(t) in lieu of vf, for instantaneous velocity instead of final velocity
  • vi in lieu of v0, for initial velocity
  • x(t) or x in lieu of d, for position/distance

While 1-D motion problems may be fairly straightforward, 2-d problems can be a little more math intensive- using algebra and trigonometry, and sometimes Calculus.  2-D projectile motion (also referred to as parabolic motion or parametric equations) often provide some initial conditions, and ask the student to solve for:

  • Velocity vector broken into horizontal (x) and vertical (y) components
  • Maximum height attained
  • Time to reach max height
  • Total time in the air
  • Total distance (horizontal range) of projectile
  • Landing or final impact velocity
  • Landing or final impact angle/direction.

2-D projectile motion problems typically represent one of these four types of situations:

Typical Projectile / Parabolic Motion, Launch & Land at same height Launch upward from an initial height and land below
Launch horizontally from an initial height (cliff/roof) and land below Launch horizontally from a moving object at an initial height and land below (similar to case 3)

Simplify the process and double check 2-D projectile motion and parabolic motion problems with these useful tools!

  1. Here’s a great tool to input your initial conditions for a 2-d kinematics problem and solve for a variety of variables.  It accepts initial conditions as a vector with initial angle, or as individual vector components.
  2. Calculate instantaneous velocity for any vertical motion problems involving gravity.  It will solve for the vertical velocity at a given time.  Note: when inputting the acceleration due to gravity, this program will automatically assume g to be negative.  So rather than input -9.8, just input 9.8.
  3. Calculate final velocity for any 1-D motion problem involving constant acceleration.  It will solve for the instantaneous velocity at a given time.
  4. Visualize a typical projectile motion problem and score points with this game.  Fire a projectile from a cannon and hit a target.  Using your pre-determined  horizontal range, you can position a target at a fixed distance from the initial position.  Use your mouse to drag the target into position.  Click and drag the + sign at the end of the yellow line (representing a measuring tape) to measure a particular distance.
  • Tystarwalt

    What if you have the time, distance, and angle, but not initial velocity?

    • Damon M.

      Initial velocity in the x direction is normally given, but while initial velocity in the y direction is always 0

      • Anson Han

        Not to mention- for 99% of the problems, the horizontal velocity is constant, there is no acceleration.