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Unformatted text preview: PHYS 5900 Class 18 (10/05/2009) Zi-Wei Lin Manipulating Equations Example A projectile in the x-y plane has an initial speed of v0 and angle theta0 . A) Derive its vertical position y as a function of x ; B) Derive the range in x ; C) Plot y as a function of x and find the numerical value of the range for the following initial condition: v0=10 m/s, theta0=30° (note: g=9.8 m s 2 ). A) In:= myEqns = 8 x '' @ t D == 0, y ' ' @ t D == - g, x ' @ D == v0 * Cos @ theta0 D , y ' @ D == v0 * Sin @ theta0 D , x @ D == 0, y @ D == < ; In:= DSolve @ myEqns, 8 x @ t D , y @ t D< , t D Out= :: x @ t D fi t v0 Cos @ theta0 D , y @ t D fi 1 2 I- g t 2 + 2 t v0 Sin @ theta0 DM>> In:= sol = 8 x, y < == 8 x @ t D , y @ t D< . % @@ 1 DD Out= 8 x, y < : t v0 Cos @ theta0 D , 1 2 I- g t 2 + 2 t v0 Sin @ theta0 DM> We now express y in terms of x by eliminating t : In:= ? Solve Solve @ eqns , vars D attempts to solve an equation or set of equations for the variables vars . Solve @ eqns , vars , elims D attempts to solve the equations for vars , eliminating the variables elims . Solve[eqns, vars, elims] solves eqns for vars , eliminating the variables elims In:= yxRule = Solve @ sol, y, t D Simplify Out= :: y fi - g x 2 Sec @ theta0 D 2 2 v0 2 + x Tan @ theta0 D>> We now need to replace fi by == in order to get the equation for y(x) : In:= FullForm @ % D Out//FullForm= List @ List @ Rule @ y, Plus @ Times @ Rational @- 1, 2 D , g, Power @ v0,- 2 D , Power @ x, 2 D , Power @ Sec @ theta0 D , 2 DD , Times @ x, Tan @ theta0 DDDDDD In:= Position @ % , Rule D Out= 88 1, 1, 0 << In:= yxRule @@ 1, 1 DD Out= y fi - g x 2 Sec @ theta0 D 2 2 v0 2 + x Tan @ theta0 D In:= yxRelation = Apply @ Equal, yxRule @@ 1, 1 DDD Out= y- g x 2 Sec @ theta0 D 2 2 v0 2 + x Tan @ theta0 D or In:= yxRule . 8 Rule fi Equal < Out= :: y- g x 2 Sec @ theta0 D 2 2 v0 2 + x Tan @ theta0 D>> B) we need to find the value of x at y=0 (the range): In:=...
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This note was uploaded on 04/25/2010 for the course PHYS 5900 taught by Professor Lin during the Fall '09 term at East Carolina University .
- Fall '09