 Now in the previous video Joe explained how natural philosophers came to understand the law of inertia That an object will continue moving at a constant speed and direction unless it's acted on by forces Now there's a lot to pick apart from that statement. What is a force first of all? How can we quantitatively describe the motion of an object and how is that motion going to change when the object is acted on by a force So we're going to start by showing how the motion of a Large complex object can be broken down into simpler parts. I Want you to think about throwing a water balloon up into the air The balloon goes up and down and it probably spins around and it changes its shape a little Now these are the three types of motion that it can occur in any situation From an explosion to a flower opening to a satellite traveling through the solar system To completely describe the motion of the water balloon we should specify exactly how each point of the balloon moves through space in time For example, if we choose the knot of the balloon, it's going to follow some curly path That's going to be really difficult to write down an equation for But it turns out that there is a special point associated with every object That moves along a much simpler path than any other part of the object and this point is called the center of mass It is much much simpler to describe how this center of mass moves through space and Then if we really need to we can describe how the rest of the object moves relative to this center of mass In most situations just knowing how that center of mass moves is enough And what this means is that we just have to track how a single point moves in time So we're going to quantitatively describe a single moving point and this means we need to keep track of its location in both time and space Now as a word in English time can mean a few different things it can mean a point in time such as what time is it? Or it can mean a duration as in how much time does it take me to swim 50 meters in Physics we write the symbol t to mean a point in time and the symbol delta t to mean a duration Now this delta t is just a single symbol. It's not two different symbols multiplying together It's just a single thing And we calculate delta t as the end time point minus the beginning time point Now location and space requires two or three coordinates, which we can study using vectors But to simplify the maths a bit. We're going to look at the case when location needs just one coordinate Once you're comfortable doing the maths with one coordinate, you'll be able to move on to working with more dimensions And in fact a single coordinate is enough to keep track of speed along a fixed path Such as a car on the highway a train on its tracks or an airplane on a straight flight path So to describe where a train is along a track We just have to write down where the center of mass is at each point of time X as a function of t There are a few decisions to make first We must decide where the zero point along the track is and maybe also what will be the zero point in time And then we have to decide which direction along the track is going to be positive It doesn't matter too much how you choose these things You just have to be consistent for the whole time you're working on a particular problem Imagine I record the position of my train as a function of time and when I plot this information It looks like a straight line Let's suppose the time axis tick marks are one second each and the x-axis tick marks are 10 meters Then we can calculate how fast the train is going as delta x over delta t and for example We get six meters per second I can choose a longer time interval say three seconds and see that the displacement is 18 meters So the velocity is still six meters per second In fact any time interval I choose will give me velocity equals six meters per second The velocity is constant and a graph of the velocity as a function of time is just a horizontal line Now if we think about acceleration There's no change in velocity and so the acceleration is zero Now let's look at a graph of x that is not a straight line We see from this graph that the train travels 10 kilometers in a 10-minute interval And we can calculate the average velocity for this trip as 60 kilometers per hour This is the slope of the line that joins the point at x for time equals zero to x at time equals 10 minutes If the train had had a constant velocity for those 10 minutes the x of t curve would look like that red line Now what further information does the blue curve tell us if we take a much smaller time interval at the beginning here There is no change in position. So the velocity is zero and if we do the same at a later time We see a steeper slope These slopes calculated for very small time intervals are called the instantaneous velocities The lines that just touch the curve here are called tangents to the curve and this is something you'll study more in calculus From looking at this blue curve of position as a function of time and the instantaneous velocities it implies It seems the train slowed down and stopped at a station briefly before Increasing its speed as it continued on its journey The velocity curve is drawn in green and just as slopes on the displacement graph gave us velocity Slopes on the velocity graph give us acceleration So for this example, we see that the acceleration is negative up until team one Then both velocity and acceleration are zero until time two When the train starts moving again and its acceleration is positive