Introduction to Motion and Acceleration

Finally 3 what path does the projectile follow

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Unformatted text preview: etric equations cos 1 2 We may eliminate the parameter by solving the first equation for : sin cos Thus, sin cos 1 2 cos Simplifying, This is a parabola. Therefore, the path the projectile follows is a parabolic one. Early in the history of the study of projectile motion if was believed that projectiles followed a path 9 composed of an initial rectilinear motion, followed by a second stage circular motion, and finally another rectilinear motion (see figure below). Circular This is what we believed projectile trajectories looked like Example 7 A bomber traveling at 640 mph releases a missile from 35,000 feet. The missile malfunctions upon launch and falls freely under the force of gravity. The intended target lies 10 miles ahead at the time of release. Will the missile hit its target? Solution 640 35,000 ft. Our objective is to determine if 10 miles. Since we will work in English units, we need to express distances in feet and speeds in ft/sec. 10 miles = 52,800 feet. Also, 640 mph = 938.67 ft/sec is the initial speed of the missile which is inherited from the bomber. The position vector for the missile is 1 2 938.67,0 . We have 0 Where 0 0,35000 , 0 0 16 or 938.67 35,000 16 In order to find we need to know . When the missile strikes the ground, Therefore, the time of flight is 0. 35,000 46.77 sec 16 The horizontal distance traveled during this time is 938.67 46.77 43,901.60 feet This is short by a several thousand feet of the desired 52,800 feet. The missile will miss its mark. 10 Unit Tangent, Unit Normal, and Binormal Vectors The study of more general forms of motions is greatly facilitated by the introduction of three very important vectors: the Unit Tangent, Unit Normal, and Binormal vectors. These vectors determine the intrinsic properties of curves in general and in particular provide a better framework for the theoretical study of motion. The Unit Tangent Vector Let a curve C be represented by the vector function . We learned that at points where defined by is tangent to the curve. Therefore, the vector the vector unit tangent . is a unit vector. We call it the unit tangent vector to the curve at , Example 8 Find the unit tangent vector of 0. at Solution 1,2 At √1 0 1 4 1 0 4 2 √1 1 4 √1 , , 2 √1 4 2 √1 4 , √5 √5 Verify that thi...
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This note was uploaded on 10/24/2012 for the course MAC 2313 taught by Professor Lopez during the Spring '10 term at Miami Dade College, Miami.

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