1CHAPTER 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 V0at an angle αto the horizon. At any subsequent time in its motion its speed is Vand the angle that its motion makes with the horizontal is ψ. The initial horizontal component if the velocity is V0cos 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= V0cos α= constant throughout the motion. The initial vertical component of the velocity is V0sin 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 Vsin 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&&gy-=&&7.1.1a,b α==cos0Vux>Vy-α==sin0v&7.1.2a,bxV t=0cosαyV tgt=-0122sinα7.1.3a,b The two equations 7.1.3a,bare the parametric equations to the trajectory. In vector form, these two equations could be written as a single vector equation: rVg0=+tt184.108.40.206 Note the + sign on the right hand side of equation 7.1.4. The vector gis directed downwards. The xy-equation to the trajectory is found by eliminating tbetween equations 7.1.3aand 7.1.3bto yield: yxgxV=-tancos.αα202227.1.5
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