 What does it mean to change our position over time?  It simply means that we move from one point to another in a certain amount of time.  Remember, we measure velocity in meters/second, or m/s.

 Let’s think about this! We know that position over time is measured in m/s and that position is measured in meters.  Maybe if we just use the same idea, we can get the right units for this thing called acceleration.  So, velocity is measured in m/s so if we divide by seconds, we get…

 We thought of velocity as the time it took to get from one position to another, so we will think of acceleration as:  The time it takes to get from one velocity to another!  A positive acceleration means that we are speeding up in the positive direction, while a negative acceleration means we are speeding up in the negative direction.

 We know that a constant velocity changes something's position, but how does acceleration change things…Does it change velocity and position?  YES! It changes both!  In a 2 dimensional world, we have 4 possible ways to think about velocity and acceleration: 1 where both are positive, 1 where both are negative, and 2 where they are opposites!

 Acceleration gets a blue arrow, while velocity get the red arrow  Lets talk about the examples where they are both the same, either both positive or negative  In this example, we are moving forward (positive velocity,) but every second that we are moving, we are moving forward faster! (positive acceleration)

 Now, we are moving in the negative direction (negative velocity,) but every second that we are moving, we are moving in the negative direction faster! (negative acceleration)

 Now, we are moving in the negative direction (negative velocity,) but every second that we are moving, we are speeding up in the positive direction! (positive acceleration)  Whoa, what does that mean???  If we are speeding up in the opposite direction we are moving, then we slow down (until net velocity is zero,) then we will actually move in the direction of the acceleration – positive direction in this case

 In this example, we are moving forward (positive velocity,) but every second that we are moving, we are speeding up in the negative direction (negative acceleration)  Even though our acceleration and velocity are in opposite directions, if we are speeding up in the opposite direction we are moving, then we slow down (until net velocity is zero,) then we will actually move in the direction of the acceleration – the negative direction in this case!

 Yes it was, but we still need to accomplish two goals:  What equation can we use to find the final position of a moving object with a constant acceleration?  We already know an equation for moving under constant velocity:  x = x o + v(t) where x is the final position, x o is the final position, v is velocity and t is time.

 Don’t forget how we figured it out!!!  We know that velocity is measured in m/s, so if we want to convert to position (just meters,) all we need to do is multiply by a time (whichever time we want to use) to turn it into a position!  Maybe we should try this again for acceleration?  YES!!!

 We already have our first equation: x = x o + v(t)  Now we just need to add in a part that deals with acceleration!  How can we turn acceleration into a position?  Well, we already know how to turn a velocity into a position, so we just need to find a way to turn acceleration into a velocity!

 No, unfortunately it is not quite that easy.  It actually took a long time to figure out why that didn’t work.  The reason comes from one of the most complicated branches of math – Calculus.  Some of you will be lucky (or unlucky, depending on how much you like math,) enough to study Calculus in high school!

 Yes, after we finish with this powerpoint, you will do some simulations to help you cement your understanding.  First, we still need to figure out how to solve for time in our equation.  This is harder than you might think!  Let’s do an example problem on the next slide

 Let’s say that we start at a position of 5 m, we end up at 10 m, out initial velocity is 2 m/s, and our acceleration is 2 m/s 2. At what time do we arrive?  A  B  C  D  E. 1

 A  B  C  D  E. 1  YES! We can’t have a negative time, so we automatically know that B is wrong!  Let’s do the problem and see which one is correct!

 A  B  C  D  E. 1  D is our correct answer!