Key Ideas:

• Projectile motion is an example of 2-D motion.

• The vertical component of the motion is accelerated by gravity

• The horizontal component of the motion has no acceleration

• The horizontal and vertical components are tied together by the time, so this is good practice with parametric equations.

Textbook Sections:‍ ‍OpenStax College Physics 2e section 3.4

Projectile Motion:

Just about every physics textbook in the world teaches vectors in two dimensions and then — immediately — jumps into projectile motion. There’s a reason for that. It is an easily accessible “case study” of the ideas that we have been looking at, and it is very tangible. We have all thrown a ball at some point, and the motion of that ball is projectile motion. However, as always, there’s a bit going on behind the scenes that we need to look at before we begin.

Simplifying Assumptions:

• The earth is flat: Obviously, this is not really true, but for any of the sort of projectiles that we will be looking at — a thrown baseball, for example — the distances involved will not be so large that the curvature of the earth is going to make any difference.

• There is no air resistance: Unlike the flat-earth assumption above, this one is kind of questionable. For example, a home-run batted baseball experiences significant air resistance due to its speed. The same is true of a golf ball. The math of air resistance is a little ways beyond this course.

• Ignore throws and catches: Obviously, a thrown ball starts off in someone’s hand. The hand accelerates the ball. We know this. But we’re not going to start analyzing the motion until the ball is free of the thrower’s hand. In the same way, the ball ends up being caught by someone, and that someone also (negatively) accelerates the ball. Again, we’re ending our analysis one instant before the ball is caught. If the ball hits the ground, we end our analysis one instant before it hits.

Turning it into equations:

If we put the assumptions above into practice, we discover something important:

The only acceleration is gravity, and it acts in the vertical direction.

Putting that into math, we actually get two equations:

Two quick notes:

• The unit for acceleration is m/s/s.

• Why “-g” rather than -9.8 m/s/s? Look back at the free fall page. We use g=9.8 m/s/s because we want equations that will work on any planet. “g” is positive because it’s a magnitude. If we mean “downward” — and we do, in this case — we put the negative outside of the “g” to indicate that.

Now, let’s combine the x and y accelerations above with the formulas from the previous section.

Recall that we derived the following:

We’re going to make two changes. First, we will just eliminate that third line equation. It’s a valid formula; it’s just not very useful for what we’re about to do. Second, we are going to substitute the correct values for the x and y accelerations. As we do this, remember that we’re substituting zero for the x acceleration. Any term that contains x acceleration is going to vanish because we multiplied it by zero!

When we make those changes, here’s what we get:

Notice that we have parametric equations? The horizontal and vertical values of position and velocity are all tied together by a single variable: time. Often, when solving problems involving projectile motion, you must first solve for a time when something happens, and then you can solve for the position or the velocity.

While we’re noticing things, also notice that we will, in most cases, get two answers. There will be an x-position and a y_position. There will be an x-velocity and a y-velocity. In a lot of cases, that’s all you’re going to need, but once in a while, a problem will ask for a magnitude and direction. When that happens, you will have to take your x and y components and convert them into magnitude and angle form. The techniques you need are found in section 1.7.