x = Vo*Cos(θ)*t
y = Vo*Sin(θ)*t + .5*g*t^2
t = 2*Vo*Sin(θ)/g
x = 2*Vo^2*Cos(θ)*Sin(θ)/g
x' = (2 * Vo^2)(cos^2(θ) - Sin^2(θ))
x' = (2 * Vo^2)(1 - 2 * Sin^2(θ))
0 = (1 - 2 * Sin^2(θ))
θ = 45 degrees
F = α * v
a = α * v / m
a = Β * v
d2y/dt2 = g + Β*dy/dt
d2y/dt2 - Β*dy/dt = g
y = C1 *e^(Βt) + C2*t*e^(Βt)
y(0) = 0
y'(0) = Vo*Sin(θ)
0 = C1 *e^(Β*0) + C2*0*e^(Β*0)
C1 = 0
y = C2*t*e^(Βt)
y' = C2*(t*Β*e^(Β*t) + e^(Βt))
C2 = Vo*Sin(θ)
Yh = Vo*Cos(θ)*t*e^(Β*t)
d2y/dt2 - Β*dy/dt = g
Yp = A*t
Yp' = A
Yp'' = 0
d2y/dt2 - Β*dy/dt = g
0 - Β*A = g
A = -g/Β
Yp = -g*t/B
y = Yp + Yh
[B]y = -g*t/B + Vo*Sin(θ)*t*e^(B*t)[/B]
d2x/dt2 = -Β*dx/dt
d2x/dt2 + Β*dx/dt= 0
x = C1 *e^(-Βt) + C2*t*e^(-Βt)
x(0) = 0
x'(0) = Vo*Cos(θ)
C1 = 0
C2 = Vo*Cos(θ)
x = Vo*Cos(θ)*t*e^(-B*t)
[B][SIZE="4"]y = -g*t/B + Vo*Sin(θ)*t*e^(B*t)
x = Vo*Cos(θ)*t*e^(-B*t)[/SIZE][/B]
0 = -g*t/B + Vo*Sin(θ)*t*e^(B*t)
t = ln(g/(B*Vo*Sin(θ)))/B
x = Vo * Cos(θ) * ln(g/(B*Vo*Sin(θ)))/B * e^(-ln(g/(B*Vo*Sin(θ))))
x' = -((e^-ln[(g Csc[θ])/(B*Vo)]*Vo*Cos[θ]*Cot[θ])/B) + (e^-ln[(g Csc[θ])/(B*Vo)]*Vo*Cos[θ]*Cot[θ]*ln[e]*ln[(g*Csc[θ])/(B*Vo)])/B - (e^-ln[(g Csc[θ])/(B*Vo)]*Vo*ln[(g*Csc[θ])/(B*Vo)]*Sin[θ])/B
0 = Cos[θ]*Cot[θ] + Cos[θ]*Cot[θ]*ln[(g*Csc[θ])/(B*Vo)]) - ln[(g*Csc[θ])/(B*Vo)]*Sin[θ]
Thats interesting, I never knew that :)The force of air resistance starts out approximately linearly (as swraman used), and is a good approx for small-medium object at low-medium speeds. For larger objects and higher speeds the resistance goes approximately as a (~)square of the velocity (i totally can't remember the details by i think the exact value of the exponent varies with the shape of the object??), and finally begins to level out asymptotically (this comes from fluid approximations where a fairly large enveloped develops around the projectile shielding it from air resistance).