If a particle moves in only one dimension, like a train on as straight track, it’s easy to answer the question about where it is because there’s only one component to it: “13m North” or “-3.6 feet.” It’s a single number with units and a positive or negative sign… that’s it. Pretty simple, right?

Well, the truth is that most objects move in two or three dimensions, and so we need more information to tell us where that object is, so we use vectors. We’re going to focus on objects moving in a two dimensional plane.

We’re going to study particles (or projectiles) that move in two dimensions. This can be a cannon ball after being fired, a baseball after being thrown, a golf ball after being hit, a soccer ball after being kicked, or any other situation you can think of where an object is under the influence of only gravity after the initial force applied to move the object. (Usually we ignore wind resistance when we do these types of problems.)

The FBD of projectiles is simply a downward pointing arrow to indicate the weight. If it looks strange to have a force not in the direction of the object’s travel path, just remember that a force isn’t needed to sustain motion… it’s actually the opposite! Objects stop moving because of the forces applied to it. The FBD are always a snapshot of the forces acting on the object in that moment. The object can be moving in one direction and the force acting in another.

A projectile is a particle that is only experiencing gravity, and in most cases, gravity is only acting in one direction. Gravity doesn’t influence the horizontal motion (if we accounted for air resistance, then there would be a force in this direction as well), only the vertical motion. That’s why the ball falls to the ground when you throw it.

This means that a bullet fired horizontally from a gun experiences a constant horizontal velocity and a downward vertical acceleration. A bullet fired from a gun pointed up at a 45 degree angle also experiences a constant horizontal velocity and a downward vertical acceleration. A bullet fired from a gun in outer space away from any gravitational influences would travel up at a 45 degree path away from the gun and experience constant horizontal and vertical velocity.

Now let’s do a set of physics problems so you can really see how to solve these. The first one shows you how to not only calculate an angle buried in a trig function, but also that you don’t need fancy equations to solve a problem and that you really have to understand what the problem is asking for, so you don’t waste time calculating stuff you don’t need.

This problem will show you how a soccer ball can also be a projectile, and how by knowing a couple of simple things, you can find out everything you need about the problem, including how far and how high the ball traveled in addition to its time of flight.

This is a simple, fun, and sneaky way of throwing tiny objects. It’s from one of our spy-kit projects. Just remember, keep it under-cover. Here’s what you need:

• a cheap mechanical pencil
• two rubber bands
• a razor with adult help

Click here to go to next lesson on Pirate Problem.

Okay now, back to work! Here’s a neat problem involving a pirate ship and a cannon ball. I seriously doubt pirates would be able to calculate this kind of problem when being fired at by a fortress, but you might have a captain that had a good sense based on experience of how far and fast that cannon ball could travel.

When you drop a ball, it falls 16 feet the first second you release it. If you throw the ball horizontally, it will also fall 16 feet in the first second, even though it is moving horizontally… it moves both away from you and down toward the ground. Think about a bullet shot horizontally. It travels a lot faster than you can throw (about 2,000 feet each second). But it will still fall 16 feet during that first second. Gravity pulls on all objects (like the ball and the bullet) the same way, no matter how fast they go.

What if you shoot the bullet faster and faster? Gravity will still pull it down 16 feet during the first second, but remember that the surface of the Earth is round. Can you imagine how fast we’d need to shoot the bullet so that when the bullet falls 16 feet in one second, the Earth curves away from the bullet at the same rate of 16 feet each second?

Answer: that bullet needs to travel nearly 5 miles per second. (This is also how satellites stay in orbit – going just fast enough to keep from falling inward and not too fast that they fly out of orbit.)

Catapults are a nifty way to fire things both vertically and horizontally, so you can get a better feel for how objects fly through the air. Notice when you launch how the balls always fall at the same rate – about 16 feet in the first second.  What about the energy involved?

When you fire a ball through the air, it moves both vertically and horizontally (up and out). When you toss it upwards, you store the (moving) kinetic energy as potential energy, which transfers back to kinetic when it comes whizzing back down. If you throw it only outwards, the energy is completely lost due to friction.

The higher you pitch a ball upwards, the more energy you store in it. Instead of breaking our arms trying to toss balls into the air, let’s make a simple machine that will do it for us. This catapult uses elastic kinetic energy stored in the rubber band to launch the ball skyward.

Click here to go to next lesson on Two Body Problems.

So now let’s look ahead and sneak a peek into your future. Are you nervous about taking Calculus? Or if you have, have you wondered what Calculus could possibly be useful for? Here’s a two part video that shows you what Calculus is (and will even have you doing it before the end of the second video!) and how it’s used all the time in physics. Sir Isaac Newton was so frustrated that he couldn’t solve his physics problems with the math that was already developed at that time (algebra) that he set them aside to invent a branch of mathematics that could support his work in science, and that’s where Calculus came from. Here’s how we use it today in physics…

Now that you know what functions are, here’s how to solve the rabbit problem:

This experiment is for Advanced Students. For ages, people have been hurling rocks, sticks, and other objects through the air. The trebuchet came around during the Middle Ages as a way to break through the massive defenses of castles and cities. It’s basically a gigantic sling that uses a lever arm to quickly speed up the rocks before letting go. A trebuchet is typically more accurate than a catapult, and won’t knock your kid’s teeth out while they try to load it.

Trebuchets are really levers in action. You’ll find a fulcrum carefully positioned so that a small motion near the weight transforms into a huge swinging motion near the sling. Some mis-named trebuchets are really ‘torsion engines’, and you can tell the difference because the torsion engine uses the energy stored in twisted rope or twine (or animal sinew) to launch objects, whereas true trebuchets use heavy counterweights.

Click here to go to next lesson on Helpful Hints.