 In this video we'll discuss projectile motion and by separating out the horizontal and the vertical components of that motion we'll derive the acceleration, the velocity and the displacement of an object undergoing projectile motion. Projectile motion occurs when a moving object is subject to a constant gravitational force and no other forces are acting on the object. This is the type of motion that basically any object travelling through the air will undergo. This is motion we all intuitively understand, even our furry friends, at least when a treat is on the line. We know that if we throw an object up into the air, it will rise continuously to a maximum height, then fall continuously until reaching a surface or the ground. We know that if we throw the same object at a steeper or shallower angle from the horizontal and we want the object to reach the same location, then we can throw the object at a different speed to affect its motion. Of course if I want to reach the greatest height at the same speed, I know that I need to throw the same object straight upwards. To experience projectile motion, we also don't need to throw the object upwards. The object still undergoes projectile motion if it's thrown at an angle below the horizontal. And if I'm feeling very mean to my diligent assistant, I can even drop the object onto the ground and we still have the conditions for projectile motion. The only force acting is gravity. Of course in this, and in most projectile motion, problems will neglect air resistance. So, when the object leaves my hand, what other force is acting on it? Well, from the definition of projectile motion, the only force acting on the object is the force due to gravity. So if we want the acceleration, it's pretty simple. That's just the acceleration due to gravity, and we know it acts downwards. We also know that there are no horizontal acceleration components. Since there's no acceleration in the horizontal direction and all the acceleration is in the vertical direction, it helps to split the motion up into vertical and horizontal components. So let's consider our vertical throw. Our acceleration, we know is just the acceleration due to gravity. To find our velocity, we can use the fact that our acceleration is equal to the change in velocity over the change in time. So for time t, we know that our change in velocity is equal to g times t. We also know that our change in velocity is equal to our final velocity minus our initial velocity. We can arbitrarily choose our final time to measure, so we can see that at any moment in time, our change in velocity is equal to the velocity at that specific time minus our initial velocity. Rearranging our equations, we can find that our velocity at time t is equal to gt plus initial velocity. So what is our vertical displacement? To find this, we can relate our velocity with our displacement. So v is equal to the change in displacement over the change in time. However, we know that our velocity is dependent on t. It is not a constant. However, we do know that our velocity has a constant acceleration, and so because our velocity is constantly increasing, we can look at the average between the velocity at two moments in time. If our initial velocity is zero and our final velocity is gt, then we know that our average velocity will be equal to gt on two. So to find our displacement at time t, we multiply this by t, and that gives us our displacement of gt squared on two. This equation is one of the classical equations of motion that you may have met previously with the acceleration fixed as small g. So what happens when our velocity is zero? Well, this is the case where we drop the treat straight onto the floor. But what about when we throw the treat at an angle? Well, we know that our horizontal and vertical axes are independent, so in the vertical direction, it doesn't matter what is happening in the horizontal direction because it won't affect our vertical motion. So am I saying that we already have the equations for our vertical motion? Exactly. So if we label our vertical direction our y-axis and relabel our variables, then we have the equations of motion in the vertical direction. However, if we want to understand our object's motion, then we still need the equations for our horizontal direction. We'll label this our x-axis. As we've discussed, there are no horizontal forces, therefore we immediately know that our horizontal acceleration is zero, and if our velocity isn't changing because our acceleration is zero, then the only possible velocity in the horizontal direction is our initial velocity, and this has no dependence on time. So our horizontal displacement is just equal to our initial velocity times time plus any initial horizontal displacement. As previously, if we're finding displacement from an initial position of zero, then this term will be zero, and our horizontal displacement will just be equal to the initial horizontal velocity times time. We now have all the equations that fully describe projectile motion, but there's a catch. If you are watching closely, you may be wondering whether theta angle has disappeared too. Well, usually we consider the speed and the angle at which an object is thrown, and not the initial vertical and horizontal velocities. We can use vector components to find our horizontal and vertical components of velocity. If our angle theta is to the horizontal, then our initial vertical velocity will be the initial velocity multiplied by sine theta, and our initial horizontal velocity will be the initial velocity times cos theta. So the final trick to doing projectile motion problems is to use algebra. A useful trick is to remember that the time t appears in both the vertical and horizontal equations. So if we want to link our vertical and horizontal results, one method is to make t the subject of our equations. You'll see how this works in the following examples.