1.  In this packet we will look at:  The meaning of acceleration  How acceleration is related to velocity and time  2 distinct types acceleration  A simple equation which help us calculate average acceleration

2.  In everyday conversations we often hear people say something along the lines of, “That car accelerates from 0 to 60 in 3.2 seconds,” or, “That runner accelerated like a cheetah.”  But what exactly do those statements mean?  What they are saying is that some object is changing its speed or velocity over some period of time.

3. Now if you have gone through some of my packets you might notice that I give or try to hint that you should always define something as precisely as possible. In the case of acceleration, this is no different. Definition: Acceleration : 1. A vector quantity which measures how an objects velocity changes with time. 2. The measure of the rate of change of an objects velocity. Note: Notice how we are concerned with change in velocity and not change in speed. Let’s look at an example to shed some light on the meaning of this definition. Example 1: A race car accelerates from 0 to 60 mph in 3.2 seconds. 3.2 seconds go by Car at rest (i.e. velocity = 0 mph) Car moving (i.e. velocity = 60 mph)

4. In physics when we see this we think, “In 3.2 seconds the velocity of the car changed from 0 mph to 60 mph.” But since most science and math people like to express things in terms of equations we try to do the same here. We write out the definition of acceleration as follows: But what’s the point of an equation when it takes as long to write out as the sentence? So we simplify the above equation to: Now we need to explain what ∆v and ∆t mean in this equation.  ∆v represents the difference between an objects final velocity and an objects initial velocity. We write it as ∆v = (v f – v i )  Likewise ∆t represents the time interval between the final and initial velocity of an object. We write it as ∆t = (t f – t i ). However, most often than not t i is 0, so we just take ∆t to be the time given in the problem.

5. Remember I mentioned that acceleration was a vector quantity? But in Example 1 there was no hint of a direction. So how can acceleration be a vector when we supposedly only have a magnitude? To understand this we need to become very good friends with the coordinate plane (e.g. the xy-plane). In fact, physics uses the concepts present in mathematical graphing to help define directions for objects in motion. Let’s take a look at Example 1 again. So the car was at rest and then 3.2 seconds later it was traveling at 60 mph. Now the car must have had to move for this to happen. Even though we don’t have a specified direction that the car moved in we know for sure that it had to move in some direction. Often times when this occurs (and it happens a lot ) we use a simple xy-plane to “imagine” for ourselves a direction an object is moving in. x+ y+ AB

6. In the case of Example 1 we can say that the car might have moved from some point A to some point B in the 3.2 seconds, then the arrow from A to B would represent our direction. This is a very subtle yet very important part of any physics problem involving vectors. The student should try to understand this idea as best they can. Let’s look at some common representations of directions. x+ y+ x- y- Movement of an object to the right and upward is usually considered positive (+). Movement of an object to the left and down is usually considered negative(-). Movement of an object to the left and up is usually considered negative(-) x motion and positive (+) y motion. Movement of an object to the right and down is usually considered positive (+) x motion and negative(-) y motion.

7. There are many more ways you can orient the xy-plane to fit your problem (e.g. through rotation of the xy-plane, horizontally flipping the xy-plane, vertically flipping the xy-plane). Understanding when to do this requires that you become comfortable with this idea and practice it. For now this basic orientation will be enough for most introductory physics problems. With this we can now see that even when we are not necessarily given a direction, we can still define one for ourselves. Now let’s move on to understanding another common term we use when talking about acceleration. In particular, often say that when a object is slowing down, that it is decelerating. Rule: Whenever we think of an object decelerating we can say that the objects acceleration is in the opposite direction of the objects motion. This idea can be very useful when we want to determine the sign of the acceleration (i.e. whether the acceleration is positive or negative).

8. Following the rule from the previous slide, let’s look at some examples of when acceleration is considered positive or negative. Velocity Acceleration Velocity Acceleration y+ x+ y+ x+ Case 1 Case 2 In Case 1 the velocity is in the positive x-direction and the object is speeding up. Since the velocity is pointing in the positive x-direction we say that the acceleration is positive. In Case 2 the velocity is in the negative x-direction and the object is slowing down. Since the velocity is pointing in the negative x-direction the acceleration is positive.

9. Velocity Acceleration Velocity Acceleration y+ x+ y+ x+ Case 3 Case 4 In Case 3 the velocity is in the positive x-direction and the object is slowing down. Since the velocity is pointing in the positive x-direction we say that the acceleration is negative. In Case 4 the velocity is in the negative x-direction and the object is speeding up. Since the velocity is pointing in the negative x-direction the acceleration is negative. Notice that Cases 1-4 are observations which you can make about objects in motion. The arrows representing the acceleration vectors are not necessarily needed to determine the sign for the acceleration of the object. What you do need is the direction of the velocity and the behavior of the object (i.e. is the object speeding up or slowing down). The acceleration arrows were added just to help the student visualize what was happening.

10. So we have defined acceleration, in words and as an equation, talked about its meaning and the sign convention that is follows. So we must be done. Well, not quite. Remember at the beginning of the slideshow I mentioned that we were going to take a look at 2 types of acceleration? We will do this by looking at the equation for acceleration we used before and start to refine it. Now we know that acceleration is a vector so we need to follow the sign conventions which vectors follow. That is we need to make acceleration boldfaced. Before we do this though, let’s talk a little about Average Acceleration and Instantaneous Acceleration. The way we have been talking about acceleration, we consider it to be a change in velocity over some finite, or measurable, change in time. This would be what Average Acceleration is. Now if we were to reduce that change in time to be infinitely small, too small to measure. We would then be able to find the magnitude of the acceleration of an object at any instant in time. This is what is know as Instantaneous Acceleration. For this packet we will focus on Average Acceleration.

11. Here the bar over acceleration represents average. Also notice that since acceleration and velocity are vectors, they are both boldfaced. This is by convention and I include it so that the student may become familiar with the convention. We can expand this equation if we like into: Notice that if you were to solve this equation for acceleration, the units for acceleration would be (units of velocity) divided by (unit of time). You could have: (mph)/s, (km/hr)/hr, (m/s)/hr, etc. But the most commonly used units are (m/s)/s or m/s 2 and (ft/s)/s or ft/s 2. The key when you want to simplify your units as much as possible is to convert your time and velocity to the same units of time.

12. Example 2: On a trip to the racetrack you notice that a horse accelerated from 0 to 30 mph in 2 seconds. What was the horses acceleration? Example 3: You and a friend start racing over a 1000 meter distance. You both run at 5 m/s until you get to the 500 meter mark. You notice that at 500 meters your friend displayed a burst of speed as she started to distance herself from you. At the finish line you found out your friend accelerated to 7.5 m/s as she crossed the finish line. You also found out that your friend finished the race in 150 seconds. What was your friends acceleration from the 500 meter mark to the 1000 meter mark? Example 4: You are driving a car on the highway, and you decided to slow down so that you can get off at an upcoming exit. You were originally travelling at 25 m/s and then decelerated at a rate of 10 m/s 2 to get off at the ramp travelling at 15 m/s. How long did it take you to slow down?

13. The first thing you should always do when doing physics problems is: 1.Write down what you are being asked to find. 2.Write down the knowns and unknowns. 3.Write down governing equation(s) for this particular problem. 4.Pick your governing equation(s) depending on what you know and what your are being asked to solve. Example 2: We are being asked to find the horses acceleration. Knowns Unknowns Governing Equation Plugging in and solving we see that the horse increased its velocity by 15 mph every second for 2 seconds.

14. Example 3: We are being asked to find your friends acceleration from the 500 meter to the 1000 meter mark. Knowns Unknowns Governing Equations This problem is a little tricky as you have three unknowns. But you have two equations at your disposal. Remember physics builds upon itself, often times you will have to use what you’ve already learned to help you solve new problems. In this problem we first need to figure out how long your friend took to get to the half way point and then how long she took to get to the finish line. Rearranging the first governing equation and solving for time we get:

15. Solving for the time it takes to get to the halfway point we get: This information is important because it tells us how much of the total race time we can ignore, since we are only interested in the race from the 500 meter mark to the 1000 meter mark. So since we know that the race took your friend 150 seconds and she spent 100 seconds the finish the first half then it must have taken here 50 seconds to finish the second half. See the tricky part here? The final time in the known column cannot be used it this case because it considers the whole race, but we are only interested in half of the race. So now t f should be 50 s and we will consider t i to be 0 s. Plugging these values into our equation for acceleration we get:

16. Example 4: We are being asked to find the time that it took to slow down. Knowns Unknowns Governing Equations Here we don’t know how long it took us to slow down, what we can take into consideration though it that we might consider our initial time to be 0 seconds and solve for our final time, since the difference in time is how long it will take to slow down. Setting t i equal to 0 and solving our governing equation for t f we get our answer:

17. But wait a minute, we cannot have a negative time. That is not physically possible. That is a good observation. Now remember how we defined acceleration be positive and negative when we relate it to velocity and when something is speed up or slowing down. See Slides 8 and 9. In this case the car, even if it is slowing down, will being moving in some direction. Let’s say we define that direction as positive. Now because the car is slowing down, then the acceleration has to be in the opposite direction. Thus our acceleration is negative. Making this change in our solution we get the following which does make sense and is correct.