Constant Acceleration NotesLevel 1: Constant Acceleration EquationsObjectivesBy the end of this section, you should be able to:

1. Read a story problem, and pull out relevant information, including “hidden” information, such as objects starting at rest, stopping…

2. Read a story problem, and identify the variable you are looking for.3. Use reasoning skills to select the appropriate equation to solve the problem.4. Solve a constant acceleration problem.

In the last unit, we looked at displacement, velocity, and acceleration. In this unit, we are going to look at equations that relate these variables together. We’ll look at rockets speeding up toward space, boxes falling off roofs, and cars screeching to a stop. In all these situations, we are going to be able to calculate a lot of information about how far these object move, how long they are in motion, their rates of acceleration….Before we can do that, we need to makes sure we know the variables that are involved in these equations.

Symbol

What it stands for Units

a Acceleration m/s/s or m/s2

d Displacement mt Time svi Initial velocity m/svf Final velocity m/s

A big part of being able to do these problems is being able to pull the variables out of the scenario. If you were given a problem, could you figure out what information was given, and what you are looking for? Try the example problem below. You don’t need to solve the problem. Just see if you can pull everything out.

Example ProblemTwo friends are riding on a sled. They start from rest, and end up moving 11 m/s. As they speed up, they move 44 m. How long did this motion take? You do not need to solve this. Just write down what you know and what you are looking for.What you know…

What you are looking for?

Okay… now that we’ve got that down, I’m going to introduce you to a few very important equations…Brace yourself. They are unspeakably beautiful.

The Constant Acceleration EquationsEquation a t d vi vf

a=v f−vit

d=v i t+a t 2

2

v f2=v i

2+2ad

d=(v f+v i ) t2

Ooooh. Purty. These are the constant acceleration equations. They can help you calculate tons of stuff. In fact, when NASA sent people to the moon, they did it based on the equations above. There is a catch, however. You can only use these equations if the acceleration is constant. You can only use these if the acceleration is steady. Fortunately, in this class, the acceleration will be constant 99.9% of the time. It should be pretty obvious when it is not.

Notice I gave you these equations in a chart. When you first start working with these, the chart makes it a lot easier to pick the correct equation to solve a problem with. Notice the dark areas on the chart. These indicate what variable is not in the equation, which turns out to be pretty useful information. Before we move on, you might want to copy the chart above onto your notecard. This will make your life A LOT easier.Read the example problem below carefully. I will show you how to use this. Take out a separate sheet a paper and follow along with the steps.

Example ProblemA car starts from rest and speeds up with an acceleration of +8.5 m/s/s. It accelerates at this rate for 5 seconds. How far does the car travel during this time?Solution

1. Write out what you know, what you don’t know, and what you are looking for. Whenever you start these, write out all the variables like this:a=¿t=¿d=¿vi=¿v f=¿

Then, take all the information you can out of the problem and fill it in. If you don’t know something and you aren’t solving for it, cross that variable out. Notice I say cross not scribble. You want to be able to see this later. It’s helpful.

a=¿8.5 m/s/st=¿5 sd=??? What we are going to solve forvi=0m /s Because the car starts from restv f=¿ Not given in the problem, not what we are solving for

2. Choose an equation. Okay, since we now have our list of variables, we can use it to choose the equation. Use the variable you crossed off to pick (the one you aren’t solving for, that isn’t given in the problem.

a=¿8.5 m/s/st=¿5 sd=??? What we are going to solve forvi=0m /s Because the car starts from restv f=¿ Not given in the problem, not what we are solving for

Equation a t d vi vf

a=v f−vit

d=v i t+a t 2

2

v f2=v i

2+2ad

d=(v f+v i ) t2

3. WRITE THE EQUATIONAt this point, your paper should look like this:

a=¿8.5 m/s/st=¿5 sd=??? vi=0m /s v f=¿

d=v i t+a t 2

2

4. Plug in the numbers and solvea=¿8.5 m/s/st=¿5 sd=??? vi=0m /s v f=¿

d=v i t+a t 2

2

d= (0 ) (5 )+ (8.5 ) (5 )2

2

d= (0 )+ (8.5 ) (5 )2

2

d= (8.5 ) (5 )2

2

d=(8.5 )(25)

2

d=106.3

5. If you like it then you better put some units on it.d=106.3 m

Now, try a problem on your own, when you finish, check the solution below. It will show you what your paper should look like.Example ProblemAn airplane lands on a runway going 85 m/s. It travels 340 m before coming to a stop. What is its acceleration while coming to a stop?SolutionWhen you finish your paper should look like:

a=¿???t=¿d=340m vi=85m /s v f=¿ 0 m/s

v f2=v i

2+2ad02=852+2a(340)0=7225+2a(340)0=7225+680a−7225=680a

−7225680

=a

−10.6 ms2

=a

This practice is long (you will see it in a second). This is because these only get easier with practice. Slow down, take your time, and help one another. Some people have had many years of algebra practice. Others have had only a year or two. Be kind and patient. Explain your thinking.

Level 2: Positives and Negatives in AccelerationBy the time you finish this level, you should be able to:

1. Identify which direction an object is moving, when given its velocity2. Use the acceleration of the object to determine whether the object is speeding

up or slowing down3. When given the velocity of an object over time, be able to identify the

acceleration.4. When given the initial velocity and acceleration, be able to predict future

velocitiesRemember that acceleration is the rate a velocity is changing. For example, if we had an object who’s motion was changing like this:Time (s) Velocity (m/s)1 302 203 304 40

The acceleration is -10 m/s/s. This is because the object’s velocity is changing by -10 m/s every second.Also remember that we can show direction using positives and negatives. A positive velocity means an object is moving in the positive direction. A negative velocity means the object is moving in the negative direction. The batmobile below is moving the negative direction.

Check for UnderstandingWhich car is moving faster, the car going -2 m/s or a car going -13 m/s?

If you said the car going -13 m/s was going faster, you were right! Negative and positives tell us what direction the object is going. The number tells us how fast it is moving.

The acceleration tells us how much a velocity is changing. Use the charts below to figure out the acceleration of the object.Time (s) Velocity (m/s)1 42 83 164 205 24

Acceleration= Acceleration= Speeding up or slowing down? Speeding up or slowing down?

In these charts the acceleration is given below. Fill in the rest of the chart.Time (s) Velocity (m/s)1 1002345

Acceleration= -10 m/s/s Acceleration= +4m/s/s Speeding up or slowing down? Speeding up or slowing down?

You may have noticed a pattern that explains whether the object is speeding up or slowing down.Key Idea:If the velocity and the acceleration have the same sign, the object is speeding up. If the velocity and the acceleration have opposite signs, the object is slowing down.

Example ProblemA car has a velocity of -13 m/s and an acceleration of +2 m/s/s. Which direction is the car moving? Is the car speeding up or slowing down?

Example ProblemA flamingo is sauntering -2 m/s toward a lake. It is accelerating -1 m/s/s. Is the flamingo speeding up or slowing down?

Time (s) Velocity (m/s)1 -252 -203 -154 -105 -5

Time (s) Velocity (m/s)1 22345

Level 3: Acceleration due to GravityObjectivesBy the time you finish this section, you should be able to:

1. Describe an object’s motion as it falls.2. Draw a picture of an object falling straight down.3. Explain why we don’t often think objects fall this way.4. Describe air resistance.5. Use the concepts above to make sense of completely new

situations.6. Calculate time, displacement… for objects in free fall.

Have a group member stand up, hold his or her pencil up and drop it. Watch the motion carefully. Does it seem like the pencil is dropping at a constant velocity or accelerating? The picture on the left was made by a camera taking pictures at a stead rate as a ball fell- click, click, click… What does this tell you about how objects fall? Why do the images of the ball get farther and farther apart?Falling objects are accelerating. They start off going slowly, then faster, and faster and faster…In fact, falling objects accelerate at the same rate, -9.8 m/s2. It doesn’t matter how big the object is or how heavy it is. The earth will pull them down with the same rate of acceleration.

This can be a hard thing for your brain to accept. Children (and many adults) believe that lighter objects will fall slower. Why? Because on earth we have air that will slow down an object’s acceleration. Lighter objects are often more affected by air than heavier objects. This is why a feather will take longer to hit the ground than a brick. They are both pulled down toward the earth at the same rate, the paper is just slowed down by the air around it.We call this air resistance. Air resistance occurs when air slows an object’s motion down. You experienced air resistance a lot as a kid when you stuck your hand out the car window.If an object is only experiencing gravity, we say it is in free fall. This means we are going to ignore air resistance. You can’t do this for everything. For example, if you are dropping a piece of paper on earth, air resistance is obviously going to effect it. It will drift around, slowed down by the air. A

brick, however, isn’t really effected by the air, so we can pretty much say it is in free fall.On earth, objects in free fall speed up with an acceleration of -9.8 m/s/s. Take a look at how the object speeds up as it falls. It starts from rest. After falling one second, it is moving -9.8 m/s. After two seconds, it has sped up to -19.6 m/s…Notice that I haven’t talked about how big the object is or how much mass it has. It doesn’t matter. All objects, regardless of how much they weigh, will fall to the ground while speeding up at the same rate. Okay, now that we have that idea down, let’s talk about direction. In physics, down is negative, up is positive. That means that the object on the right is actually speeding up in the negative direction. To help yourself remember this, change all the velocities on the right to negatives. They are negative because the object is moving down.If you need help remembering this, imagine hopping off the edge of a cliff. Falling down would be a negative experience. Finding you can fly up would be a positive experience.Because of this, we say all objects fall at a rate of -9.8 m/s/s.

Falling ProblemsThis is fun because whenever we have an object in free fall, we automatically know its acceleration. If the object starts from rest, we also know the object’s initial velocity is zero.Most students love these problems, once they get the hang of them, because they are fairly easy. You just have to learn to look for the “hidden” information in the problem. Let’s try one together.

Example ProblemKing Kong is at rest when he falls from the top of the Empire State Building (which is 443 m tall). How long does he have to think about life and betrayal on the way down? Try it yourself first, then see how I solved it.

SolutionKing Kong is at rest when he falls from the top of the Empire State Building (which is 443 m tall). How long does he have to think about life and betrayal on the way down?

d=−443m Negative because the object is moving downt=?a= -9.8 m/s/s Because they are on earthvi= 0 m/s Because King Kong starts from restvf

d=v i t+a t 2

2

−443=(0)t+(−9.8) t2

2

−443=(−9.8) t 2

2

−443=−4.9t 2

−443−4.9

=t 2

90=t 2

√90=√t 2

9.49 s=t