Question

Find x(t), y(t), z(t), v_x(t), v_y(t), v_z(t) for a projectile in 3-D with air resistance and wind resistance.

Given: vector for v_o=(v_x,v_y,v_z), vector for wind v_w=wŷ

Equation of motion of projectile: F_tot=ma=-my(z-hat)- β(vector for v_o)+ β(vector for v_w)y

Answer 1

... (1)

Subst,

we get, comparing the x, y, z part of the differential eqn (1),

...(2)

...(3)

...(4)

(2), (3) and (4) are the linear ordinary differential equations.

Solving (2),

...(5)

Solving (3),

...(6)

Solving (4),

...(7)

where C₁, C₂, C₃ are constants which can be evaluated if initial conditions are given.

Now, solving (5),

Similarly, solving (6) we get,

Solving (7),

Here D₁ , D₂ , D₃ are constants which can be evaluated given initial conditions.