2D Motion Problem Solving

The big thing we learned from the lab we did last time is that horizontal and vertical motion are completely independent. That means that a marble that is rolling off the edge of a table will take the same amount of time to hit the ground as one that is dropped straight down. Gravity doesn’t care at all about the horizontal motion of an object, it just pulls them all down the same way. Once again, all objects fall with the same constant acceleration.

That makes our problem solving very easy, since we already know how to solve freefall problems. Now all we have to do is consider horizontal motion as well, but that’s really easy. Since gravity doesn’t affect things horizontally, the horizontal motion is constant speed, with zero acceleration. So solving a two-dimensional motion problem is like solving a freefall problem plus a constant speed problem. Piece of cake.

Let’s try one:

If I climb a tree that is 15 m tall and throw a ball horizontally at 12 m/s, how far from the trunk of the tree will the ball land?

The key point in setting up these problems is to keep your horizontal and vertical information separate. Make two tables for what you know, and draw a line between them.

vertical / horizontal

Note that the horizontal speed given in the problem is an average speed. Since the horizontal speed is not changing, there is no initial speed or final speed, only average speed. That also means that the only equation from our toolbox that you’ll need for horizontal calculations is the first one, .

Also, there’s really no such thing as horizontal or vertical time. Time is just time. So if you can use the information you have to solve for the time, you get to write that number in the vertical and horizontal boxes. On the vertical side, we have three things we know, so we can solve for the other two, particularly time.

Now that we know how much time the ball is going to be in the air, we can figure out how far it will travel horizontally. Since it’s moving horizontally at 12 m/s for just under 2 seconds, we know that our answer should be something less than 24 m. Using the first equation in our toolbox:

Here’s another one to try:

A bowling ball rolls horizontally off a cliff at 6 m/s and lands 20 m from the base of the cliff. How tall is the cliff?

Again we make our table, listing the things we know, keeping vertical and horizontal information separate.

vertical / horizontal

Now in this problem, we don’t have enough information to solve the vertical part yet. But if we knew how much time the ball was in the air, we could solve it. So we use the horizontal information to figure out the time.

The ball is in the air for 3.3 seconds. Using our freefall shortcut this time (since initial velocity is zero),

Key points:

  • Keep the vertical and horizontal information separate.
  • Any horizontal speeds in the problem are average speeds, not initial or final speeds.
  • Time is the only thing that crosses over between horizontal and vertical.