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  ODEs: Basic Methods   PHYS 416/517   Garcia, Chapter 2    Projectile Motion   The Basic Equations for 2D projectile motion are   Where   (1)   d  ! rdt  = ! vd  ! vdt  =  ! g  ˆ  y + 1 m ! F  a ( ! v ) m  = m ass  (kg) g  = acceleration due to gravity ( ! 9.81 m/s 2 ) ! r  =  position (  x ,  y ) ! v  =  velocity (  x ,  y ) ! F  a ( ! v ) = any additional force such as air resistance  Projectile motion - no air resistance   The solution for the projectile motion in a vacuum is:    x ( t  ) =  x 0  + v 0 t  cos !   y (  y ) =  y 0  + v 0 t  sin !   !  12  gt  2 where:(  x 0 ,  y 0 ) ! initial position v 0  ! initial speed !   –initial launch angle g  –acceleration due to gravity  Projectile motion - no air resistance   In the simple case of an object launched with initial speed v 0  and an angle with respect to the ground at a height y 0, one can show that the time of flight t   R  is   the range is:   and the maximum height is:   t   R = v 0 sin !  +  v 02 sin 2 !  +  2 gy 0 g    x max =  x 0 +  v 0 t   R  cos !     y max =  y 0 + v o 2 sin 2 !  2 g ! 
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