Explanation:
(A) Sum of forces on the projectile in the y direction:
-bv − mg = ma
Acceleration is the derivative of velocity with respect to time:
-bv − mg = m dv/dt
Separate the variables:
bv + mg = -m dv/dt
-1/m dt = 1/(bv + mg) dv
-b/m dt = b/(bv + mg) dv
Integrate:
-b/m t |₀ᵗ = ln(bv + mg) |₀ᵛ
-b/m (t − 0) = ln(bv + mg) − ln(0 + mg)
-b/m t = ln(bv + mg) − ln(mg)
-b/m t = ln((bv + mg) / mg)
e^(-b/m t) = (bv + mg) / mg
bv + mg = mg e^(-b/m t)
bv = -mg + mg e^(-b/m t)
v = -mg/b (1 − e^(-b/m t))
Velocity is derivative of position with respect to time:
dz/dt = -mg/b (1 − e^(-b/m t))
Separate the variables:
-b/(mg) dz = (1 − e^(-b/m t)) dt
Integrate:
-b/(mg) z |ᵧᶻ = (t + m/b e^(-b/m t)) |₀ᵗ
-b/(mg) (z − h) = (t + m/b e^(-b/m t)) − (0 + m/b e^(0))
-b/(mg) (z − h) = t + m/b e^(-b/m t) − m/b
z − h = -mg/b (t + m/b e^(-b/m t) − m/b)
z = h − mg/b (t + m/b e^(-b/m t) − m/b)
(B) Repeat steps from part A, but this time in the x direction.
-bv = ma
-bv = m dv/dt
-b/m dt = 1/v dv
-b/m t |₀ᵗ = ln v |ᵥᵛ
-b/m (t − 0) = ln vₓ − ln v₀ₓ
-b/m t = ln (vₓ / v₀ₓ)
vₓ / v₀ₓ = e^(-b/m t)
vₓ = v₀ₓ e^(-b/m t)
dx/dt = v₀ₓ e^(-b/m t)
dx = v₀ₓ e^(-b/m t) dt
x |₀ˣ = -m/b v₀ₓ e^(-b/m t) |₀ᵗ
x − 0 = -m/b v₀ₓ e^(-b/m t) − (-m/b v₀ₓ e^(0))
x = -m/b v₀ₓ e^(-b/m t) + m/b v₀ₓ
x = m/b v₀ₓ (1 − e^(-b/m t))
To find z(x), find t in terms of x then substitute into z(t).
b x / (m v₀ₓ) = 1 − e^(-b/m t)
e^(-b/m t) = 1 − b x / (m v₀ₓ)
-b/m t = ln(1 − b x / (m v₀ₓ))
t = -m/b ln(1 − b x / (m v₀ₓ))
z = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) + m/b (1 − b x / (m v₀ₓ)) − m/b)
z = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) + m/b − x / v₀ₓ − m/b)
z = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ)
The range is when z = 0:
0 = h − mg/b (-m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ)
h = mg/b (-m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ)
bh/(mg) = -m/b ln(1 − b x / (m v₀ₓ)) − x / v₀ₓ
-(b/m)² h/g = ln(1 − (b/m) x / v₀ₓ) + (b/m) x / v₀ₓ
Unfortunately, this can't be simplified further without using something called the Lambert W function.
(C) The range of a projectile without air resistance launched horizontally from a height h is:
x = v₀ₓ √(2h/g)
