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0LID?=NCF? -INCIH

by Mike Maloney

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What is a Projectile? Here are 4 examples of projectile motion. Let’s find

out what is similar and different about them.

Drop Vertical Toss

Roll off Table 2-D throw

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What is a Projectile? A projectile is an object upon which the only force

acting is gravity.

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What is a Projectile? A projectile is an object upon which the only force

acting is gravity.

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What is a Projectile? A Free-Body Diagram of a Projectile would

look like this. It would have one force vector acting on it, and its

direction would be in the direction gravity is acting.

Fg

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What is a Projectile? To illustrate this, let us use an example of a cannon

horizontally shooting a projectile. What would the path of the cannon ball be without

any forces or gravity acting upon it?

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What is a Projectile? To illustrate this, let us use an example of a cannon

horizontally shooting a projectile. What would the path of the cannon ball be without

any forces or gravity acting upon it?

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What is a Projectile? Now, what would the path look like with the force

due to gravity added?

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What is a Projectile? Gravity acts downward on the ball and causes it to

have a parabolic trajectory. It only affects its vertical motion.

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A Projectile’s Trajectory Projectiles can be analyzed in two dimensions. This means that there is an X (horizontal) and Y

(vertical) component of a projectile’s motion

VERTICAL (FREE FALL) HORIZONTAL (CONSTANT V)

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A Projectile’s Trajectory

These components act independent of each other Gravity accelerates the ball downwards, but cannot

affect its horizontal motion which remains constant.

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A Projectile’s Trajectory Let’s consider an example of two balls falling off a

table to illustrate this. One will roll of the edge, one will just be dropped

… which one will hit the ground first?

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A Projectile’s Trajectory The object’s trajectory is defined when the effect of

gravity is combined with the effect of the object’s initial velocity.

It is accelerated vertically, but moves at a constant velocity horizontally.

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A Projectile’s Trajectory Now let’s introduce a few terms that describe

the projectiles path.

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A Projectile’s Trajectory The range is the horizontal distance that a

projectile moves.

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A Projectile’s Trajectory The max height is the highest point that a

projectile reaches during its flight.

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A Projectile’s Trajectory The launch angle (θ) is the angle above the

horizontal that a projectile is launched at.

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Conclusion The force of gravity accelerates projectiles downward.

This downward force and acceleration result in a downward displacement from the straight-line motion that the object would travel if there were no gravity.

The force of gravity does not affect the horizontal component of motion. Since there are no other forces acting upon it, a projectile maintains a constant horizontal velocity.

The vertical and horizontal components of a projectile operate independent of each other, and their combined effect is the addition of their separate effects.

Due to this, projectiles travel with a parabolic trajectory. That trajectory can be described in terms of its range,

max height, and launch angle.

Paraphra

se!!!

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Quick EvaluationThinking about the cannon example that was just discussed, which diagram could represent the ball’s ...

initial horizontal velocity initial vertical velocity horizontal acceleration vertical acceleration net force Initial velocity

A B or C none B B A, B, C, D, E

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Quick Evaluation

Suppose a zookeeper wanted to shoot a banana into a monkey’s mouth as the monkey was falling out of a tree. Where should the zookeeper aim the banana to get it into the monkey’s mouth. Above, At, or Below the monkey?

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The Math of Projectile Motion In order to understand the

motion of a projectile, it must be thought of as a vector.

Just as any other vector, it can be thought of as having two perpendicular components.

It has a horizontal component and a vertical component.

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The Math of Projectile Motion

To describe projectiles, first let us think back to 1-D Kinematics.

What three different terms do we use to describe the motion of an object?

That’s right: Acceleration Velocity Displacement (or position)

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The Math of Projectile Motion

For our discussion let’s use a few conventions to make things more simple.

We will denote horizontal motion by the subscript “x”

We will denote vertical motion by the subscript “y”

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The Math of Projectile Motion

X-direction(horizontal)

Acceleration = ax Velocity = vx Displacement = x

Y-direction(vertical)

Acceleration = ay Velocity = vy Displacement = y

This means that the motion of a projectile can be described by the following variables…

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What is happening to a Projectile

In both cases its velocity changes in the y-direction but remains constant in the x.

Let’s see if we can create some equations that describe this.

Jump to ExamplesJump to Examples

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Acceleration of a Projectile

What is a projectile’s acceleration in the x-direction?

ax = 0

How about the y-direction? ay = -9.8 m/s2

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Velocity of a Projectile

What equation describes the instantaneous velocity of an object?

Right: v = v0 + a*t

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v = v0 + a*t

Now we need a couple more variables.

Lets call vxo this horizontal component of the projectile’s initial velocity (v0).

And we can call vyo the vertical component of the initial velocity (v0).

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v = v0 + a*t

What would a projectile’s velocity in the x-direction be?

ax = 0 vx = vxo

How about the y-direction?ay = g = - 9.8 m/s2

vy = vyo + (g)*t

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Displacement of a Projectile

All we have left is the displacement.

What equation describes the displacement of an object that is undergoing acceleration?

Right: x = v0*t + ½ a*t2

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∆d = vi*t + ½ a*t2

So, what is a projectile’s displacement in the x-direction? ax = 0, so the second term is 0.

x = vxo * t

How about the y-direction? y = vyo*t + ½ g*t2

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Projectile Equations

ax = 0 ay = ag (on Earth = 9.8 m/s2) vx = vxo vy = vyo + ag·t x = vxo·t y = vyo·t + ½ ag·t2

These equations can be used to solve any projectile problems. With a little algebra, you can solve for any of the variables.

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Determining Vxo and Vyo A few more things, and our description of

projectile motion will be complete. We have to find out how Vxo and Vyo relate to V0. Remember, the initial velocity is a vector, with a

magnitude and a direction (angle). So we can use VECTOR RESOLUTION to help

deal with this problem. V0 can be broken into components just as

any other vector can be.

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V0 can be broken into components that are perpendicular to each other, and in the x and y directions we are using.

V0

Vxo

Vyo

Determining Vxo and Vyo

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Using our old friends Sine and Cosine, we can determine the magnitude of Vxo and Vyoin terms of V0.

V0

Vxo

Vyo

Determining Vxo and Vyo

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Vxo is adjacent to the angle θcos θ =Vxo / V0So, Vxo = V0·cos θ.

Vo

Vxo

Vyo

= V0·cos θ

Determining Vxo and Vyo

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Vyo is opposite the angle θsin θ =Vyo / VoSo, Vyo = Vo·sin θ.

V0

Vxo

Vyo= V0·sin θ

Determining Vxo and Vyo

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Projectile Equations

ax = 0 ay = ag (9.8 m/s2) vx = vxo vy = vyo + ag·t x = vxo·t y = vyo·t + ½ ag·t2

• Vxo = V0·cos(θ)• Vyo = V0·sin(θ)

So in the end, we have …

V0

Vxo

Vyo

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Projectile Motion Let’s see how this all comes together. Here is an animation that shows the change in a

projectile’s velocity during its flight As you can see, it moves the same distance

horizontally every second. And falls farther each second than it did the second

before. It maintains at a constant speed in the x-direction

throughout the flight And is always accelerated downward (slows down on

the way up, and speeds up on the way down)