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Vertical and Horizontal Projectile Notes Level 1: Vertical Projectiles We are now entering a new phase of kinematics. So far, we’ve discussed a variety of methods for analyzing motion in general: definitions (displacement), for example), equations (constant acceleration equations, for example), and graphs. In this section, we are going to start exploring projectiles. Projectiles are object which is only being influenced by gravity. They are all around us. Toss your pen up in the air. From the time it leaves your hand until the time it returns, it is a projectile. For this unit, we will ignore the effects of air resistance. Over the next few weeks, we are going to look at three different types of projectiles. 1. Vertical Projectiles We’ve already looked at one type of these when we did free fall problems. Vertical projectiles are projectiles that are are falling down or have been tossed straight up in the air. Because we are already pretty familiar with some of these, this is where we’ll start. 2. Horizontal Projectiles We will then look at projectiles that are launched horizontally off a surface. This includes text books tossed out of windows, cars driving off cliffs, or children running off the end of piers. They look like this:
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Vertical and Horizontal Projectile NotesLevel 1: Vertical Projectiles

We are now entering a new phase of kinematics. So far, we’ve discussed a variety of methods for analyzing motion in general: definitions (displacement), for example), equations (constant acceleration equations, for example), and graphs.

In this section, we are going to start exploring projectiles. Projectiles are object which is only being influenced by gravity. They are all around us. Toss your pen up in the air. From the time it leaves your hand until the time it returns, it is a projectile. For this unit, we will ignore the effects of air resistance.

Over the next few weeks, we are going to look at three different types of projectiles.

1. Vertical Projectiles

We’ve already looked at one type of these when we did free fall problems. Vertical projectiles are projectiles that are are falling down or have been tossed straight up in the air. Because we are already pretty familiar with some of these, this is where we’ll start.

2. Horizontal Projectiles

We will then look at projectiles that are launched horizontally off a surface. This includes text books tossed out of windows, cars driving off cliffs, or children running off the end of piers. They look like this:

3. Projectiles at an angle

We will look at this in a later unit. These are the most complicated but also the most fun. This will help us calculate objects shot up in the air at an angle, including cannons fired, golf balls hit, and catapult rocks fired at castles. They look like this:

Don’t worry if this looks complicated. We will take it a step at a time.

Vertical Projectiles

You are already familiar with some of these. Vertical projectiles are object that have been dropped, thrown down toward the ground or thrown straight up in the air. The most important thing to remember is that all projectiles on earth have a vertical acceleration of -9.8 m/s/s/

What does this mean? If you thrown an apple straight up, it has an acceleration of -9.8 m/s/s. If you drop a pencil, it has an acceleration of -9.8 m/s/s. If you Spike a football toward the ground the acceleration is -9.8 m/s/s.

Key Idea

For all vertical projectiles on earth, a=-9.8 m/s/s.

For all of these, you are going to be using the constant acceleration equations. Remember these? Here they are:

Equation

a

t

d

vi

vf

The one way these can get tricky is keeping track of the initial velocity. Remember, in physics up is positive, down is negative.

Example Problem

For each of the scenarios below, decide whether the initial velocity is positive, negative or zero.

A ball is tossed up in the air.

An earring falls to the ground.

A football is spiked down toward the ground.

Using this thinking, try the practice problem below.

Practice Problem

A man takes a cell phone and throws it on the ground. The cell phone leaves the man’s hand going 3.4 m/s downward. When it hits the ground, it is moving 5.6 m/s downward. How long was the cell phone in the air?

If you did this correctly, you should have found the cellphone fell for 0.22 s.

Objects Thrown Up

If you throw an object up in the air, it has a positive velocity (up) and a negative acceleration (-9.8 m/s/s). This means the object is slowing down. It looks like this:

For one split second, at the very top, the ball comes to a stop before speeding up in the opposite direction. We call this point the apogee. The apogee is the highest point a projectile reaches.

Key Idea

At apogee, an object’s vertical velocity is zero.

Knowing that the velocity is zero at apogee is actually incredibly useful. For example, you can now solve a problem like the one below.

Example Problem

A baby is tossed up in the air with a velocity of 2.6 m/s. How high will the baby go?

Solution

Let’s first figure out our variables.

-9.8 m/s/s

Not given in the problem, not what we are solving for

What we are going to solve for

Because the baby stops at apogee

Plug n’ chug:

Coming Back Down

What happens to a vertical projectile on the way down? It reverses its motion, starting from rest (at apogee) and speeding up at a rate of -9.8 m/s/s. At each height, it will have the same velocity coming down as it did going up.

This means that if you throw a ball straight up in the air, it will come down and hit your hand going the exact same speed it was going when it left your hand. If you tossed it up going 4.5 m/s, it will hit your hand going -4.5 m/s.

That means that if you know how long a projectile takes to reach apogee, you can easily find the total time it was in the air (hang time). Just multiple by two.

This is also why its stupid to fire a gun straight up in the air. If the bullet leaves your gun going 220 m/s, it will return to that height going -220 m/s. If it hits someone, it will essentially be like getting hit point blank range. Don’t do it. Physics says no.

Level 2: Review Acceleration and Constant Velocity

There are no notes for this section. You can skip to the practice.

Level 3: Horizontal Projectiles

There is a reason that I had to you do constant velocity problems and constant acceleration problems in the last section. We will use these equations to solve problems involving horizontal projectiles.

What is a Horizontal Projectile?

Imagine you ran off a diving board and there was no gravity. What would happen to you? You would keep moving forward (with a horizontal velocity) at a steady pace, not accelerating downward.

This picture shows what would happen to a cannon ball if there was no gravity acting on it.

Because there is nothing to slow it down or speed it up, the cannonball would just keep going at the same velocity.

On the other hand, if we just dropped the cannon ball off the cliff with no horizontal velocity, it would accelerate straight down. It wouldn’t stay the same velocity- it would move faster and faster as it moved downward.

A horizontal projectile is a projectile that starts off a horizontal velocity and no vertical velocity. That sounds complicated but it isn’t. Imagine yourself running off the end of a diving board. The exact second you run off the board, you are moving forward with a horizontal velocity. You haven’t started speeding up downward yet, so you have not vertical velocity.

Figure 1: All horizontal motion. No vertical motion (yet).

In horizontal projectiles, two types of motion are happening at exactly the same time. In the horizontal direction, there is nothing to slow you down, so you keep moving at a constant velocity. In the vertical direction, you are accelerating downward at a -9.8 m/s/s.

Horizontal Direction

Vertical Direction

Constant Velocity

Accelerating

Look very carefully at the picture below to help you make sense of what we just discussed:

A projectile is doing two things at once- it is moving forward at a constant velocity, and it is accelerating downward at -9.8 m/s/s. When these two motions are combined they make a parabola, which is the shape of the path the projectile takes. See that curve above? Parabola.

This sounds complicated, but we can make it a lot easier by making a chart like this:

Horizontal

Vertical

Use:

Use:

Constant Acceleration equations

This chart is definitely note card worthy. Before you start any horizontal projectile problems, I want you to copy down the chart above (everything except the part where I wrote “Use: Constant Acceleration Equations”).

Level 4: Horizontal Projectile Problems

These are a lot better when you try them out in a problem. Let’s do a problem together.

Example Problem

A happy dog runs off the end of a diving board going 4.5 m/s forward. The diving board is 2 m high.

a) How long is the dog in the air before he splashes into the water?

b) How far away from the end of the diving board does the dog hit the water?

Take out a blank sheet of paper and try the problem out with me below. I will ask to see this paper before I will let you take your challenge (along with your practice, of course).

Do this now on your separate sheet.

Horizontal

Vertical

Use:

Use:

Constant Acceleration equations

Step 2: Fill in the Chart

This is actually a little harder than it sounds. First, look up at the problem and circle any information that might be useful. Now, the tricky part is deciding whether that information is something horizontal or vertical. Take a look at the picture below:

For each of these variables, decide whether this information is describing something vertical or horizontal:

Variable

Vertical or Horizontal? Neither?

m/s

high

d=?? Distance away he will hit water?

t =?? How long will he be in the air?

Don’t forget that there is also some hidden information here, too. In the vertical direction, the object is basically in free fall, so we know that and m/s/s.

Use all this information to fill in the chart below.

Horizontal

Vertical

Use:

Use:

Constant Acceleration equations

If you did this right, it should look like this:

Horizontal

Vertical

Because they move down

The question asks us how long the dog is in the air. Which side appears to have enough information to solve this? The horizontal or the vertical?

The vertical. Choose the equation you would use to solve for time. Remember, on the vertical side, you can only use constant acceleration equations. Actually write the equation in your chart on that side.

Horizontal

Vertical

Step 3: Solve for Time

Do it. I dare you.

Horizontal

Vertical

If you did this right, you should have found the time to be 0.64 s.

Here is something really important. Time is the one variable that is the same for both the horizontal and vertical direction. Time is directionless, so it is the one variable who can jump the fence and be used on the other side.

Step 4: Solve for the Rest of the Problem

The problem also asked you to find out how far the dog will land away from the diving board. Now that you have more information on the horizontal side, you can use the velocity equation to solve for the horizontal distance the dog will go as he is falling.

Horizontal

Vertical

0.64

A Quick Note

I’ve just shown you how to solve one type of horizontal projectile problem. In reality, there are lots of different kinds. I could sit here all day coming up with new ways to mix up the variables or have you solve for different information. When you are solving these, you won’t be able to memorize a process that works every time. These are more like puzzles than anything. The only way to get good at them is to practice, practice practice. It’s okay to get frustrated. It is not okay to give up.