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Projectile Motion

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What is a projectile? Any object that moves through the air or through space, acted on only by gravity A projectile is any object which once projected continues in motion by its own inertia and is influenced only by the downward force of gravity. Examples – a stone thrown into the air, a ball rolling off a cliff, a cannonball shot from a cannon.

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Some Visuals

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2 Components Horizontal Component of Motion – similar to a ball rolling freely along a level surface; when friction is negligible, a rolling ball will move at a constant velocity (equal intervals of time). If there is no horizontal force acting on the object, there is no horizontal acceleration.

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2. Vertical Component of Motion – similar to a freely falling object – acceleration due to ____________???. As the ball falls, the increasing speed causes a greater distance to be covered each second.

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Without gravity…

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With gravity… **The ball follows a “parabolic” path.**

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More Notes… The horizontal and vertical velocity components of vectors are independent of each other (have no affect) . But… the combined effects of each component produce the curved paths that projectiles follow. What do you call the sum or product of two projectile vectors?

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**The Monkey and Zookeeper**

Suppose the zookeeper must shoot the banana from the banana cannon to the monkey who hangs from the limb of a tree. This particular monkey has a habit of dropping from the tree the moment that the banana leaves the muzzle of the cannon. The zookeeper is faced with the dilemma of where to aim the banana cannon in order to hit the monkey. If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon? To ponder this question, first consider a scenario in which there is no gravity acting on either the banana or the monkey. What would be the path of the banana? Would the banana hit the monkey?

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**…with gravity, fired above**

If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon? To ponder this question, consider the scenario in which the zookeeper aims above the monkey (perhaps she presumes that gravity will accelerate the banana downward such that it hits the monkey). What would be the path of the banana? Would the banana hit the monkey?

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**The banana moves in a parabolic path in the presence of gravity**

The banana moves in a parabolic path in the presence of gravity. In the presence of gravity, the monkey also accelerates downward once he lets go of the limb. Both banana and monkey experience the same acceleration since gravity causes all objects to accelerate at the same rate regardless of their mass. Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. The banana misses the monkey, moving over his head. The banana passes as far above the monkey's head as it was originally aimed.

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**…with gravity, aimed at the monkey with a fast speed**

To ponder this question, consider the scenario in which the zookeeper aims at the monkey and shoots the banana very fast (perhaps she presumes that if shot fast enough, the banana will hit the monkey before it falls very far). What would be the path of the banana? Would the banana hit the monkey?

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**The banana moves in a parabolic path in the presence of gravity**

The banana moves in a parabolic path in the presence of gravity. In the presence of gravity, the monkey also accelerates downward once he lets go of the limb. Both banana and monkey experience the same acceleration since gravity causes all objects to accelerate at the same rate regardless of their mass. Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. Thus, the banana hits the monkey. Since the banana left the muzzle moving very fast, the banana reaches the monkey before the monkey has fallen very far.

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**…with gravity, aimed at the monkey, slowly**

If the monkey lets go of the tree the moment that the banana is fired, then where should she aim the banana cannon? To ponder this question, consider the scenario in which the zookeeper aims at the monkey, yet shoots the banana very slow. What would be the path of the banana? Would the banana hit the monkey?

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**The banana moves in a parabolic path in the presence of gravity**

The banana moves in a parabolic path in the presence of gravity. In the presence of gravity, the monkey also accelerates downward once he lets go of the limb. Both banana and monkey experience the same acceleration since gravity causes all objects to accelerate at the same rate regardless of their mass. Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. Thus, the banana hits the monkey. Since the banana left the muzzle moving very slow, the banana reaches the monkey after the monkey has fallen considerably far.

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The point is… A projectile is an object upon which the only force is gravity. Gravity acts to influence the vertical motion of the projectile, thus causing a vertical acceleration. The horizontal motion of the projectile is the result of the tendency of any object in motion to remain in motion at constant velocity. Due to the absence of horizontal forces, a projectile remains in motion with a constant horizontal velocity; horizontal forces are not required to keep a projectile moving horizontally. The only force acting upon a projectile is gravity!

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The Truck and The Ball Imagine a pickup truck moving with a constant speed along a city street. In the course of its motion, a ball is projected straight upwards by a launcher located in the bed of the truck. What will be the path of the ball and where will it be located with respect to the pickup truck?

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**As the ball falls, it also undergoes a downward acceleration.**

As can be seen from the above animation, the ball follows a parabolic path and remains directly above the launcher at all times. As the ball rises towards its peak, it undergoes a downward acceleration. An upwardly moving ball which is slowing down is said to be undergoing a downward acceleration. As the ball falls, it also undergoes a downward acceleration. A downwardly moving ball which is gaining speed is said to have a downward acceleration. This downward acceleration is attributed to the downward force of gravity which acts upon the ball. If the ball's motion can be approximated as projectile motion (that is, if the influence of air resistance can be assumed negligible), then there will be no horizontal acceleration. In the absence of horizontal forces, there would be a constant velocity in the horizontal direction. This explains why the ball would be located directly above the launcher from which it is projected. Many would insist that there is a horizontal force acting upon the ball since it has a horizontal motion. This is simply not the case. The horizontal motion of the ball is the result of its own inertia. When projected from the truck, the ball already possessed a horizontal motion, and therefore will maintain this state of horizontal motion unless acted upon by a horizontal force.

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Practice Problems 12. A plane is flying at an altitude of 8000 m at a sped of 250 m/s. At what horizontal distance ahead of its target must a water balloon be released to strike the target on the ground(neglecting air resistance)?

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