Throw a ball across a football field. Slide down a hill on a sled. How can you describe the motion in each case using physics? Throwing a ball is an example of projectile motion, while sledding on a hill is an example of motion down an inclined plane. Both are good applications of the vector equations of motion, because they are intrinsically two- or three-dimensional physics problems.

Projectile motion

Trajectory

Imagine kicking a soccer ball toward a distant goal. The ball starts moving upward at an angle. Then it moves along a curved path called a trajectory, gradually redirecting its velocity vector so it points downward again, back toward the ground. The trajectory of the soccer ball is a mathematical curve called a parabola that is the result of the fact that gravity accelerates moving objects down but not sideways.

Projectiles

A projectile is any object that is moving and affected only by gravity. If we ignore friction for a moment, the soccer ball is an example of a projectile.

Range

The range of a projectile is the horizontal distance it covers before landing on the ground at its starting height. When using standard coordinate axes x and y, the range is the distance measured in the x-direction.

Independence of motion in two dimensions

Underlying the physics of projectile motion is a powerful concept: that the motion in each direction (x and y) can be analyzed independently of the other! The equations of accelerated motion in vector form look just like they did in one-dimensional motion—except that there are separate equations for x and y!

What the subscripts mean

There are a lot of subscripts, so it is worth explaining them. In one dimension, a subscript “0” as in x_{0} means “initial position”; therefore, in vector components x_{0} means the x-component of the initial position and y_{0} means the y-component of the initial position. The variables x, v_{x}, and a_{x} refer to the time-dependent x-components of position, velocity, and acceleration. The variables y, v_{y}, and a_{y} refer to the time-dependent y-components of position, velocity, and acceleration. Time t is a scalar, so there is only one value for time.