PROJECTILE MOTION. ( ) g y 0. Equations ( ) General time of flight (TOF) General range. Angle for maximum range ("optimum angle")

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1 PROJECTILE MOTION Equations General time of fliht (TOF) T sin θ y 0 sin( θ) General rane R cos( θ) T R cos θ sin( θ) sin( θ) y 0 Anle for maximum rane ("optimum anle") θ opt atan y 0 atan v f atan v f θ opt atan y 0 asin y 0 acos y 0 y 0 Maximum rane (i.e. rane at optimum anle) R max y 0 v f TOF at optimum anle y 0 Maximum y at optimum anle y 0 y 0 (c) W. C. Evans 00

2 Final anle for optimum initial anle (max rane) y 0 θ T atan θ v opt θ T 0 also, usin neative of this anle for final anle, then the initial slope (tanent of initial anle) and final slopes are neative inverses, so vectors are perpendicular R max tan( θ opt ) y 0 tan θ T x intersect y intersect tan θ opt tan θ T x intersect R max y 0 y intersect y 0 x intersect y 0 point of intersection of these tanent lines (use neative for theta-t); y is on directrix ( x R max ) x y 0 y T tan θ T y tan θ opt equations of tanent lines Special cases zero initial heiht note max R here is half the max heiht (i.e., at 90 derees) v T : sin( θ) 0 R : sin( θ) cos ( θ ) sin ( θ ) θ maxr : atan( ) zero anle y 0 R : T R max θ X : asin X : y 0 Equations of motion: parametric usually x(0) 0 and accel in x is 0 x( : x 0 t cos( θ) a x t v x ( : cos θ y( : y 0 t sin( θ) t v y ( : sin θ t Equations of motion: y(x) as v0 becomes lare, this becomes linear (frozen rope) y( x) cos( θ) x tan( θ) x y0 dy dx ( cos( θ) ) x tan ( θ ) (c) W. C. Evans 00

3 Vertex form of parabola x maxy : sin ( θ ) cos ( θ ) y maxy : y 0 sin θ t maxy : sin ( θ ) y( x) cos( θ) x sin θ y 0 sin θ p ( cos( θ) ) y D y 0 directrix x F sin θ focus ( ) y F y 0 cos θ zero initial heiht, optimum anle (5 derees) x maxy y maxy t maxy Velocity vector v( sin θ : t α( : atan tan θ α( x) atan t x cos θ tan θ ( cos( θ) ) ( cos( θ) ) manitude anle vs. time anle vs. x Final (impac velocity v T : y 0 manitude α( T) : atan tan θ y tanent ( x) : ( x R) tan θ v y ( T) : sin θ y 0 ( ) y 0 ( cos( θ) ) y 0 ( cos( θ) ) anle tanent line equation y component (c) W. C. Evans 00 3

4 Galileo anles (for equal rane; zero initial y only) θ acos R : θ : acos R θ θ : T : tan θ T Maxima ellipse all the vertices (maxima) of a set of trajectories for iven initial velocity, and zero initial y, as anle varies to 90 derees, will fall on this ellipse x y Taretin solutions iven initial velocity and heiht (can be nonzero), and a point (X,Y), find the anle to hit the point β a β X b X c y 0 Y β X D b a c D > 0 two anles D < 0 no anles D 0 one anle ψ b D b D ψ a θ a atan ψ θ atan ψ parabola of safety Y X y 0 taret outside this envelope cannot be hit, reardless of anle X tanent Y tanent tan θ tan( θ) ( ) y 0 point where trajectory is tanent to envelope Velocity vector slope (i.e., derivative of envelope ) at tanent point tan θ Velocity initial slope tan θ so these are perpendicular T X cos θ time of fliht to taret X v( T) X tan( θ) velocity manitude at taret cos( θ) X α( T) atan tan θ velocity anle at taret ( cos( θ) ) (c) W. C. Evans 00

5 First-order air resistance (dra) solution x cos( θ) v x ( cos θ ( exp( ) exp Tmax ln sin θ ( y v y ( v 0 sin θ ( exp( ) t y 0 v 0 sin θ exp( as used here depends on mass; smaller for larer mass, smaller for smaller dra coeff; note larer mass approaches vacuum solution (all these equations do) x max sin ( θ ) cos ( θ ) sin( θ) sin θ y max y 0 ln sin θ T sin θ sin θ y 0 sin( θ) sin( θ) y 0 this TOF is approximate, only ood for small, approaches non-dra for small ; since this is not a very ood approximation, no point in doin rane, etc. Tilted terrain φ is the anle of the terrain, positive counterclocwise from horizontal at y0; bous solutions are possible with these; use carefully; watch neative x,y x Intersection ( cos( θ) ) ( tan( θ) tan( φ) ) y Intersection tan( φ) xi y 0 L x I cos φ distance alon hill vertex coordinates same as for flat terrain (φ 0) T sin( θ) tan( φ) cos( θ) ( sin( θ) tan( φ) cos ( θ )) sin θ flat terrain, same as y0 0 above for TOF θ opt φ launch anle for max downslope distance, which is L opt sin φ Line of siht anle measured positive counterclocwise from launch point; taret at (X,Y) φ atan Y y 0 X if Y 0, X R φ atan y 0 for θ 0; airplane / bomb (c) W. C. Evans 00 5