24 Projectile Motion
So far we have only discussed objects in free fall that are going up and down. However, objects in free fall can also have motion along the horizontal direction. This is often referred to as projectile motion. Let’s look at one such example.
Exercise 24.1: Kicking a Ball
You are standing on a flat field and kick the ball so it shoots out with an initial speed 
at angle 
with respect to the horizontal. How far will the ball go?
Since this is our first problem in 2D problem involving forces, we will go through it step by step.
Now our picture. I find it helpful to trace out the path of the ball, which we know goes up and down in an arc. If you’re not sure how I got my picture, I also added a video in which I go through the whole process.
Now find relations. Since this is a problem involving motion in 2D, we will need to write Newton’s laws for each object and for each axis!
From Newton’s laws, you will be able to figure out how the object is accelerating along both the 
and 
axis. Use that knowledge to determine how the ball moves along each of the two axis.
Use the relations we just derived to determine how far the ball travels.
Use your to determine at what angle you should kick the ball to achieve the longest possible kick. You could do this with calculus, but I think it is much more gratifying if you just reason it out in English.
I want to emphasize two things about our solution above:
- When solving force problems, we always need to write Newton’s law for each object, for each axis.
- Once we have our forces set, we can treat the two axis independently of each other.
Thus, when looking at the motion of an object along the -axis, you only care about the forces along the axis. Whatever is going on along the other axis is completely irrelevant.
Let’s do another projectile motion problem.
Exercise 24.2: Cannon Shot
You are on a castle at the top of a cliff edge, overlooking the ocean. You have a cannon that shoots cannon balls with a speed . You aim the cannon so it is perfectly horizontal, and fire. Where will the cannon ball land, and how fast is the cannon ball traveling when it splashes into the ocean?
Follow the 4 steps of problem solving. You can peak at my solution if you get stuck, but then try to finish up the problem on your own.
To end, let’s try a more conceptual question (no math needed!).
Exercise 24.3: Bullet Fired vs Bullet Dropped
A bullet is fired from a gun that is pointed perfectly horizontally. At the exact same time, a second bullet is dropped from the exact same height.
Fortunately for us, the Myth Busters performed this exact experiment.
This problem highlights a point we raised earlier: the kinematic equations for one axis are completely independent of what is happening in other axis. The “dropping” motion of the bullet is along the y-axis, so whatever is going on along the x-axis (i.e. the bullet being fired) is completely irrelevant.
Key Takeaways