Kinematics: Two-Dimensional Motion

Projectile Motion

So far in this course, all of the motion you’ve analyzed has been in one direction (either the x or the y). You also need to be able to describe as well as predict the motion of an object that is traveling in more than one direction (two-dimensions—on a plane). Any object’s motion could be described in two-dimensions. When treating motion in two-dimensions, you’ll specifically resolve the motion into two component motions—horizontal (x) and vertical (y).

Key concept Mathematical Models: Motion in Two-Dimensions

x-component y-component
V sub x equals v not x plus a sub x times t V sub y equals v not y plus a sub y times t
x equals x not plus v not x times t plus one half a sub x times t squared y equals y not plus v not y times t plus one half a sub y times t squared
v sub x squared equals v not x squared plus 2 a sub x times delta x v sub y squared equals v not y squared plus 2 a sub y times delta y

Notice, that these mathematical models are exactly the same as one-dimensional motion—just be careful, you must first resolve any velocity or acceleration vectors into components BEFORE using these equations.

Projectiles

One specific case of two-dimensional motion is projectile motion. So, what is a projectile? A projectile is an object that is being thrown and undergoes motion in both a horizontal direction and a vertical direction. Its acceleration is the acceleration of gravity.

Click the image below to view the animation of an object undergoing projectile motion.

View the animation.

In the animation, you noticed that the horizontal distance, or range, is defined by constant velocity. Also, the path of the two-dimensional motion of the projectile is called its trajectory.


Key concept Mathematical Models: Projectile Motion

x-component y-component
V sub x equals v not x V sub y equals v not y plus g times t
X equals x not plus v not x times t Y equals y not plus v not y times t plus one half time g times t squared
  V sub y squared equals v not y squared plus 2 times g times delta y