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Vectors and Projectiles

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A vector is a quantity which has both magnitude and direction. Examples of vectors include displacement, velocity, acceleration, and force. To fully describe one of these vector quantities, it is necessary to tell both the magnitude and the direction. For instance, if the velocity of an object were said to be 25 m/s, then the description of the object's velocity is incomplete; the object could be moving 25 m/s south, or 25 m/s north or 25 m/s southeast. To fully describe the object's velocity, both magnitude (25 m/s) and direction (e.g., south) must be stated.

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Most of us are accustomed to the following form of mathematics: 6 + 8 = 14. Yet, we are extremely uneasy about this form of mathematics: 6 + 8 = 10 and 6 + 8 = 2 and 6 + 8 = 5.

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Vectors are quantities which include a direction There are a number of methods for carrying out the addition of two (or more) vectors. The most common method is the head-to-tail method of vector addition. Using such a method, the first vector is drawn to scale in the appropriate direction. The second vector is then drawn such that its tail is positioned at the head (vector arrow) of the first vector. The sum of two such vectors is then represented by a third vector which stretches from the tail of the first vector to the head of the second vector. This third vector is known as the resultant - it is the result of adding the two vectors. The resultant is the vector sum of the two individual vectors. Of course, the actual magnitude and direction of the resultant is dependent upon the direction which the two individual vectors have.

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This principle of the head-to- tail addition of vectors is illustrated in the animation below. In each frame of the animation, a vector with magnitude of 6 (in green) is added to a vector with magnitude of 8 (in blue). The resultant is depicted by a black vector which stretches from the tail of the first vector (8 units) to the head of the second vector (6 units).

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As can be seen from this, 8 + 6 could be equal to 14, but only if the two vectors are directed in the same direction. All that can be said for certain is that 8 + 6 can add up to a vector with a maximum magnitude of 14 and a minimum magnitude of 2. The maximum is obtained when the two vectors are directed in the same direction. The minimum s obtained when the two vectors are directed in the opposite direction.

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Vector Addition: The Order Does NOT Matter

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The question often arises as to the importance of the order in which the vectors are added. For instance, if five vectors are added - let's call them vectors A, B, C, D and E - then will a different resultant be obtained if a different order of addition is used. Will A + B + C + D + E yield the same result as C + B + A + D + E or D + E + A + B + C?

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the order in which two or more vectors are added does not effect the outcome. Adding A + B + C + D + E yields the same result as adding C + B + A + D + E or D + E + A + B + C! The resultant, shown as the green vector, has the same magnitude and direction regardless of the order in which the five individual vectors are added.

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The Plane and The Wind

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Each plane is heading south with a speed of 100 mi/hr. Each plane flies amidst a wind which blows at 20 mi/hr. In the first case, the plane encounters a tailwind (from behind) of 20 mi/hr. The combined effect of the tailwind and the plane speed provide a resultant velocity of 120 mi/hr. In the second case, the plane encounters a headwind (from the front) of 20 mi/hr. The combined effect of the headwind and the plane speed provide a resultant velocity of 80 mi/hr. In the third case, the plane encounters a crosswind (from the side) of 20 mi/hr. The combined effect of the headwind and the plane speed provide a resultant velocity of 102 mi/hr (directed at an 11.3 degree angle east of south). These three resultant velocities can be determined using simple rules of vector addition. In the case of the crosswind, the Pythagorean Theorem and SOH CAH TOA are utilized to determine the magnitude and the direction of the resultant velocity.

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The River Boat

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Assuming that in each case the motor of the boat propels it across the river with the same force, in which case (with or without a current) will the boat make it across the shore the soonest? You might be surprised by your observation.

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Parabolic Motion of Projectiles A projectile is an object upon which the only force is gravity. Gravity, being a downward force, causes a projectile to accelerate in the downward direction. The force of gravity could never alter the horizontal velocity of an object since perpendicular components of motion are independent of each other. A vertical force does not effect a horizontal motion. The result of a vertical force acting upon a horizontally moving object is to cause the object to deviate from its otherwise linear path.

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The Monkey and Zookeeper There is an interesting monkey down at the zoo. The monkey spends most of its day hanging from a limb of a tree. The zookeeper feeds the monkey by shooting bananas from a banana cannon to the monkey in the 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?

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If there were no gravity, then what would happen if the banana was shot at the monkey? What path would the banana take and would it hit the monkey?

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Gravity-Free Environment

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Aiming Above the Monkey If there was gravity acting upon both the monkey and the banana (the usual situation), then what would happen if the banana was shot above the monkey? What paths would the banana and the monkey take? Would the banana fall (below the straight-line path) and hit the monkey as the monkey drops from the tree? Or would the banana miss the monkey, passing over his head?

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Aiming at the Monkey - Fast If there was gravity acting upon both the monkey and the banana (the usual situation), then what would happen if the banana was thrown at the monkey with a fast speed? What paths would the banana and the monkey take? Would the banana fall (below the straight-line path) and hit the monkey as the monkey drops from the tree? Or would the banana merely move in a straight line and hit the monkey immediately? Or would the banana miss the monkey, passing over his head (or even below his head)?

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Aiming at the Monkey - Slow If there was gravity acting upon both the monkey and the banana (the usual situation), then what would happen if the banana was thrown at the monkey with a slow speed? What paths would the banana and the monkey take? Would the banana fall (below the straight-line path) and hit the monkey as the monkey drops from the tree? Or would the banana not have enough speed to hit the monkey as it was falling (presumably, because the monkey would fall faster than the slow banana)?

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Horizontally Launched Projectiles Imagine a cannonball being launched from a cannon atop a very high cliff. Imagine as well that the cannonball does not encounter a significant amount of air resistance. What will be the path of the cannonball and how can the motion of the cannonball be described?

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Non-Horizontally Launched Projectiles Imagine a cannonball being launched at an angle from a cannon atop a very high cliff. Imagine as well that the cannonball does not encounter a significant amount of air resistance. What would be the path of the cannonball and how could the motion of the cannonball be described? The animation below depicts such a situation. The path of the cannonball is shown; additionally, the horizontal and vertical velocity components are represented by arrows in the animation.

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Maximum Range Imagine a cannonball launched from a cannon at three different launch angles - 30-degrees, 45- degrees, and 60-degrees. The launch speed is held constant; only the angle is changed. Imagine as well that the cannonballs do not encounter a significant amount of air resistance. How will the trajectories of the three cannonballs compare? Which cannonball will have the greatest range? Which cannonball will reach the highest peak height before falling? Which cannonball will reach the ground first?

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The Plane and The Package Consider a plane moving with a constant speed at an elevated height above the Earth's surface. In the course of its flight, the plane drops a package from its luggage compartment. What will be the path of the package and where will it be with respect to the plane? And how can the motion of the package be described? The animation below depicts such a situation.

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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. Imagine as well that the ball does not encounter a significant amount of air resistance. What will be the path of the ball and where will it be located with respect to the pickup truck? How can the motion of the ball be described? And where will the ball land with respect to the truck?

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Satellite Motion A satellite is often thought of as being a projectile which is orbiting the Earth. But how can a projectile orbit the Earth? Doesn't a projectile accelerate towards the Earth under the influence of gravity? And as such, wouldn't any projectile ultimately fall towards the Earth and collide with the Earth, thus ceasing its orbit?

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These are all good questions and represent stumbling blocks for many students of physics. We will discuss each question here. First, an orbiting satellite is a projectile in the sense that the only force acting upon an orbiting satellite is the force of gravity. Most Earth-orbiting satellites are orbiting at a distance high above the Earth such that their motion is unaffected by forces of air resistance. Indeed, a satellite is a projectile.

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Second, a satellite is acted upon by the force of gravity and this force does accelerate it towards the Earth. In the absence of gravity a satellite would move in a straight line path tangent to the Earth. In the absence of any forces whatsoever, an object in motion (such as a satellite) would continue in motion with the same speed and in the same direction. This is the law of inertia. The force of gravity acts upon a high speed satellite to deviate its trajectory from a straight-line inertial path. Indeed, a satellite is accelerating towards the Earth due to the force of gravity.law of inertia

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Finally, a satellite does fall towards the Earth; only it never falls into the Earth. To understand this concept, we have to remind ourselves of the fact that the Earth is round; that is the Earth curves. In fact, scientists know that on average, the Earth curves approximately 5 meters downward for every 8000 meters along its horizon. I

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If you were to look out horizontally along the horizon of the Earth for 8000 meters, you would observe that the Earth curves downwards below this straight-line path a distance of 5 meters. In order for a satellite to successfully orbit the Earth, it must travel a horizontal distance of 8000 meters before falling a vertical distance of 5 meters. A horizontally launched projectile falls a vertical distance of 5 meters in its first second of motion. To avoid hitting the Earth, an orbiting projectile must be launched with a horizontal speed of 8000 m/s. When launched at this speed, the projectile will fall towards the Earth with a trajectory which matches the curvature of the Earth. As such, the projectile will fall around the Earth, always accelerating towards it under the influence of gravity, yet never colliding into it since the Earth is constantly curving at the same rate. Such a projectile is an orbiting satellite.

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To further understanding the concept of a projectile orbiting around the Earth, consider the following thought experiment. Suppose that a very powerful cannon was mounted on top of a very tall mountain. Suppose that the mountain was so tall that any object set in motion from the mountaintop would be unaffected by air drag. Suppose that several cannonballs were fired from the cannon at various speeds - say speeds of 8000 m/s, less than 8000 m/s, and more than 8000 m/s. A cannonball launched with speeds less than 8000 m/s would eventually fall to the Earth. A cannonball launched with a speed of 8000 m/s would orbit the Earth in a circular path. Finally, a cannonball launched with a speed greater than 8000 m/s would orbit the Earth in an elliptical path.

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Launch Speed less than 8000 m/sProjectile falls to Earth

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Launch Speed equal to 8000 m/sProjectile orbits Earth - Circular Path

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Launch Speed greater than 8000 m/sProjectile orbits Earth - Elliptical Path

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Two final notes should be made about satellite motion. First, the 8000 m/s figure used in the above discussion applies to satellites launched from heights just above Earth's surface. Since gravitational influences decrease with the height above the Earth, the orbital speed required for a circular orbit is less than 8000 m/s at significantly greater heights above Earth's surface.

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Second, there is an upper limit on the orbital speed of a satellite. If launched with too great of a speed, a projectile will escape Earth's gravitational influences and continue in motion without actually orbiting the Earth. Such a projectile will continue in motion until influenced by the gravitational influences of other celestial bodies.

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