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CLASS7 - 1 CHAPTER 7 PROJECTILES 7.1 No Air Resistance We...

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1 CHAPTER 7 PROJECTILES 7.1 No Air Resistance We suppose that a particle is projected from a point O at the origin of a coordinate system, the y -axis being vertical and the x -axis directed along the ground. The particle is projected in the xy -plane, with initial speed V 0 at an angle α to the horizon. At any subsequent time in its motion its speed is V and the angle that its motion makes with the horizontal is ψ . The initial horizontal component if the velocity is V 0 cos α , and, in the absence of air resistance, this horizontal component remains constant throughout the motion. I shall also refer to this constant horizontal component of the velocity as u . I.e. u = V 0 cos α = constant throughout the motion. The initial vertical component of the velocity is V 0 sin α , but the vertical component of the motion is decelerated at a constant rate g . At a later time during the motion, the vertical component of the velocity is V sin ψ , which I shall also refer to as v . In the following, I write in the left hand column the horizontal component of the equation of motion and the first and second time integrals; in the right hand column I do the same for the vertical component. Horizontal. Vertical 0 = x & & g y - = & & 7.1.1 a , b α = = cos 0 V u x & gt V y - α = = sin 0 v & 7.1.2 a , b x V t = 0 cos α y V t gt = - 0 1 2 2 sin α 7.1.3 a , b The two equations 7.1.3 a , b are the parametric equations to the trajectory. In vector form, these two equations could be written as a single vector equation: r V g 0 = + t t 1 2 2 . 7.1.4 Note the + sign on the right hand side of equation 7.1.4. The vector g is directed downwards. The xy -equation to the trajectory is found by eliminating t between equations 7.1.3 a and 7.1.3 b to yield: y x gx V = - tan cos . α α 2 0 2 2 2 7.1.5

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