As we now move to work with describing the behavior (kinematics) of object moving in 2 dimensions we make use of 1 fundamental idea. This idea is that, in general, the behavior of an object along two or more orthogonal (independent or perpendicular are words with nearly the same meaning) axes are independent of each other. If we can describe what an object is doing in the E/W direction and separately describe what it is doing along the N/S direction the actual behavior will be the vector combination of the individual behaviors.

This image of a ball dropped simultaneously next to a ball shot horizontally is offered as a modest demonstration that I am not making this up. The ball shot horizontally is falling at exactly the same rate (images are at the same vertical locations) as the dropped ball. This suggests that the horizontal motion is completely independent of the the vertical motion.

The initial assumption we will make in almost all projectile problems is that air drag is not a major player. IF we make this assumption then the only acceleration in any setting will be the acceleration produced by the force of gravity (often referred to as the 'suck of the earth') which is straight down and has a magnitude of 10 (m/s)/s (ok,ok, you can call it 9.8 (m/s)/s if you must but don't expect me to go there:)) Our descriptive kinematics equations (that describe the x/t and v/t graphs of objects with constant acceleration including 0) remain the same.Vector components are always relevant in projectile problems since there is usually a horizontal (x) and vertical component of the velocity at all times. We separate the problem into vertical and horizontal parts to get started and then reassemble those parts at points where we have an interest.

There are a pile of physics tutorials for projectile motion that range from the ridiculous to the sublime. There are none that are a perfect match for our approach (which is worrisome at some level) but ultimately they all represent overlapping approaches to building understanding. What would probably be more useful is to take what you think you understand from our explorations and then see if you can understand these three exploration of actual project time motion from one of my heroes Rhett Allain. They are water hose projectile motion, Angry Birds Physics, and Clicky Pen projectile motion. Have fun!

At it's roots kinematics is a process of describing a setting and anticipating or calculating other features of the setting in a self consistent way. This leads to a general approach to descriptive physics problems called a 'frame'. The way I think about these settings is captured in the Kinematics Problem Solving Frame that I passed out/will pass out in class and is linked here.

Let's explore this frame by actually addressing a real setting. Bruce will shove an object off the counter and we will measure where it lands and how high the counter is. Does your experience suggest that where the object lands on the floor depends on how fast it was going when it left the counter? What else does it depend on? Let's see where we get after that...... [follow the steps in the problem solving frame]

Here is a youtube video of a nicely done traditional solution to a projectile motion problem. Our approach will be mostly similar so I don't think you will find this misleading. As you are watching this consider the Kinematics Problem Solving Frame that I passed out in class and is linked here.

 

With some luck we'll have enough time to create a problem with an angled launch to try our skills one. If you are doing this on your own pick a launch angle and a launch velocity and see what you can figure out.

Acceleration in more than one direction:

Projectile motion is an example of a common but relatively simple system where there is only acceleration in one direction (downwards due to gravity). It is actually relatively difficult to find other settings where there is constant acceleration in different directions. We can imagine some sort of hocky puck with rockets that provide constant accelerations in the x and y direction. Very contrived but let's do it just to be sure you understand how.

Imagine this bizzare hocky puck in on a very smooth ice rink and the acceleration in the x direction is 1.0 m/s² and the acceleration in the y direction is 2.0 m/s². What happens? This will take much longer to do than to write this paragraph but the frame we use is exactly the same.

Extracting velocity and acceleration from position information:

How would you describe the position (x and y) of an object going around in a circle at a constant speed? We will not pick any particular speed or radius to practice getting away from numbers.

More realistically:

For most real settings of two dimensional motion the acceleration in one or both directions is not constant either in magnitude or in direction. This is the case with air drag. For reasons that we'll learn about in a few weeks the acceleration of an object due to the air drag depends on the square of the speed of the object. a_drag=Kv² where K is some constant. Make a sketch of an object flying horizontally off a counter where air drag must be considered (imagine this experiment taking place underwater). Draw and label your picture. Sketch the path of the object if there were no drag force. Sketch the path of the object with the drag force. Explain your reasoning to your neighbor. Sketch the velocity and acceleration vectors at a point halfway through the 'flight' of the object. What can you say about these vectors now compared to when object began it's journey? Can you sketch the v_x and v_y vrs time plots for it's motion? Are these plots straight lines? Why or why not? Do our kinematic equations apply? Is there a way to describe the acceleration or velocity as a function of time?

FEM:

The standard way to solve interesting problems like this is called Finite Element Analysis (FEM). It is an iterative numerical method that has gotten much easier since the advent of computers and coding. If you know the acceleration in all directions initially then you calculate the where the object is as it's new velocity and direction a short time later (1 ms perhaps). Then you stop and figure out the new accelerations and their new directions. From there you can calculate where the object will be another short time later as well as the new velocity and direction. Rinse and repeat until complete. Most meaningful problems turn out to require a numberical/computational approach. Consider this moment from the movie 'Hidden Figures'.

 

I'd love to include coding in this class but I suspect you're already feeling a little overwhelmed. I may yet push you this way but I want to be respectful of your sanity......