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Projectiles

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Projectile Motion Something is fired, thrown, shot, or hurled near the earth’s surface. Horizontal velocity is constant. Vertical velocity is accelerated. Air resistance is ignored.

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**1-Dimensional Projectile**

Definition: A projectile that moves in a vertical direction only, subject to acceleration by gravity. Examples: Drop something off a cliff. Throw something straight up and catch it. You calculate vertical motion only. The motion has no horizontal component.

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**2-Dimensional Projectile**

Definition: A projectile that moves both horizontally and vertically, subject to acceleration by gravity in vertical direction. Examples: Throw a softball to someone else. Fire a cannon horizontally off a cliff. Shoot a monkey with a blowgun. You calculate vertical and horizontal motion.

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3-7 Projectile Motion The speed in the x-direction is constant; in the y-direction the object moves with constant acceleration g. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly. Figure Caption: Multiple-exposure photograph showing positions of two balls at equal time intervals. One ball was dropped from rest at the same time the other was projected horizontally outward. The vertical position of each ball is seen to be the same at each instant.

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**Horizontal Component of Velocity**

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**Horizontal Component of Velocity**

Newton's 1st Law Is constant Not accelerated Not influence by gravity Follows equation: x = vixt

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**Vertical Component of Velocity**

Newton's 2nd Law Undergoes accelerated motion Accelerated by gravity (9.8 m/s2 down) Big 3 Kinematic equations apply: vy = viy gt y = yi + viyt 1/2gt2 vy2 = viy2 2g(y – yi)

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**Horizontal and Vertical**

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**THREE TYPES OF PROJECTILES**

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**Launch angle Definition: The angle at which a projectile is launched.**

The launch angle determines what the trajectory of the projectile will be. Launch angles can range from -90o (throwing something straight down) to +90o (throwing something straight up) and everything in between.

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**Zero Launch Angle Projectiles**

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Zero Launch angle vo A zero launch angle implies a perfectly horizontal launch.

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**Examples of Projectile Motion**

Launching a Cannon ball

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**General Launch Angle Projectiles**

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**vo General launch angle **

Projectile motion is more complicated when the launch angle is not straight up or down (90o or –90o), or perfectly horizontal (0o).

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What we will need? We will need to be given the direction of the two components which we will be asked to find. In this situation, we will find the horizontal and the vertical components (velocity, acceleration, or displacement for this chapter.)

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**Speed 5.6 Projectiles Launched at an Angle**

Without air resistance, a projectile will reach maximum height in the same time it takes to fall from that height to the ground. The deceleration due to gravity going up is the same as the acceleration due to gravity coming down. The projectile hits the ground with the same speed it had when it was projected upward from the ground.

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**5.6 Projectiles Launched at an Angle**

Without air resistance, the speed lost while the cannonball is going up equals the speed gained while it is coming down. The time to go up equals the time to come down.

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**think! 5.6 Projectiles Launched at an Angle**

A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration?

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**think! 5.6 Projectiles Launched at an Angle**

A projectile is launched at an angle into the air. Neglecting air resistance, what is its vertical acceleration? Its horizontal acceleration? Answer: Its vertical acceleration is g because the force of gravity is downward. Its horizontal acceleration is zero because no horizontal force acts on it.

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**think! 5.6 Projectiles Launched at an Angle**

At what point in its path does a projectile have minimum speed? Answer: The minimum speed of a projectile occurs at the top of its path. If it is launched vertically, its speed at the top is zero. If it is projected at an angle, the vertical component of velocity is still zero at the top, leaving only the horizontal component.

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**PROJECTILE MOTION IS PARABOLIC**

That means y is a function of x an has the form y = ax - bx2, where a & b are constants for any specific parabolic motion. RELATIVE VELOCITY--The main reason we need vectors!

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**Trajectory of a 2-D Projectile**

x y Definition: The trajectory is the path traveled by any projectile. It is plotted on an x-y graph.

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**Trajectory of a 2-D Projectile**

x y Mathematically, the path is defined by a parabola.

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**Trajectory of a 2-D Projectile**

x y For a projectile launched over level ground, the symmetry is apparent.

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**Range of a 2-D Projectile**

x y Range Definition: The RANGE of the projectile is how far it travels horizontally.

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**Maximum height of a projectile**

y Maximum Height Range The MAXIMUM HEIGHT of the projectile occurs when it stops moving upward.

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**Maximum height of a projectile**

y Maximum Height Range The vertical velocity component is zero at maximum height.

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**Maximum height of a projectile**

y Maximum Height Range For a projectile launched over level ground, the maximum height occurs halfway through the flight of the projectile.

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**Acceleration of a projectile**

x y g g g g g Acceleration points down at 9.8 m/s2 for the entire trajectory of all projectiles.

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**Velocity of a projectile**

x y v v v vo vf Velocity is tangent to the path for the entire trajectory.

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**Velocity of a projectile**

x y vx vy vx vy vx vy vx vx vy The velocity can be resolved into components all along its path.

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**Velocity of a projectile**

x y vx vy vx vy vx vy vx vx vy Notice how the vertical velocity changes while the horizontal velocity remains constant.

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**Velocity of a projectile**

x y vx vy vx vy vx vy vx vx vy Maximum speed is attained at the beginning, and again at the end, of the trajectory if the projectile is launched over level ground.

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**Velocity of a projectile**

vo - vo Launch angle is symmetric with landing angle for a projectile launched over level ground.

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**Time of flight for a projectile**

to = 0 The projectile spends half its time traveling upward…

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**Time of flight for a projectile**

to = 0 2t … and the other half traveling down.

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Things to remember! Velocity is always tangent to the path. You do not know the magnitude…but it is tangent to the path. Velocity is zero at the instance you change direction. Changing Vx Vx = 0 Changing Vy Vy = 0

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**3-8 Solving Problems Involving Projectile Motion**

Conceptual Example 3-9: The wrong strategy. A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Show that he made the wrong move. (He hadn’t studied physics yet.) Ignore air resistance. Figure 3-26. Response: Both the water balloon and the boy in the tree start falling at the same instant, and in a time t they each fall the same vertical distance y = ½ gt2, much like Fig. 3–21. In the time it takes the water balloon to travel the horizontal distance d, the balloon will have the same y position as the falling boy. Splat. If the boy had stayed in the tree, he would have avoided the humiliation.

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