The Artilleryman’s Range Equations
The ballistic trajectory of a projectile follows a parabolic path defined by the following equations:
At any time t, the horizontal and vertical displacement: x and y from the origin are given by:
x = V0.t.cos(Θ)
y = V0.t.sin(Θ) - ½.g.t2
The total flight time is given by:
t = 2.V0.sin(Θ)/g
t is the time of flight
g is the acceleration due to the Earth's gravity at sea level = 9.8 m/s2
V0 is the Intial velocity
Θ is the initial angle of elevation
x is the horizontal displacement
y is the verical displacement
No aerodynamic drag
Constant g force
Launch and target both on same level
Range = V02sin2Θ/g
Maximum Range = V02/g occurs when the angle of elevation of the initial launch = 45°
Maximum Altitude During Trajectory = V02sin2Θ/2g
Maximum Possible Altitude = V02/2g which occurs when the angle of elevation of the initial launch Θ = 90° (A missile lauched straight up would fall back on the head of the launcher)
Note that the 70 and 20 degree trajectories have the same range, as do any pair of launches at complementary angles.
Missile Drag with Altitude and Speed
Drag Force (Newtons) = 0.5 x P x V2 x Cd x A
P = Density of Air (kg/m3)
The density of air decreases linearly with altitude decreasing the drag by the same ratio
P ≈ 1.29 kg/m3 @ sea level
P ≈ 0.232 kg/m3 @ 12,000 m (7.5 Miles) = 18% of the air density at sea level
P ≈ 0.001 kg/m3 @ 50 kilometers (The stratopause)
V = Velocity (m/s) or Air speed
The drag increases as the square of the velocity
Mach 1 (the speed of sound) = 340 m/s @ sea level
Mach 1 ≈ 295 m/s @ 12,000 m altitude
Cd = Co-efficient of Drag
The drag coefficient is constant for a given missile.
It depends mostly on the shape of the missile and its appendages
Cd ≈ 0.6 to 0.95 for rockets
A = Sectional Area (m2)
The effective cross sectional area is constant for a given missile
A ≈ 2.76 m2 for a 1.65 m diameter missile. (The V-2)
See more about the Components of Drag
For a wing or a lifting body at a given angle of attack the lift force is given by:
Aerodynamic Lift Force (Newtons) = 0.5 x P x V2 x CL x A
CL = Co-efficient of Lift
The lift coefficient is constant for a given angle of attack and a given wing/body shape.
It depends on the cross sectional profile of the wing or lifting body and the angle of attack.
CL ≈ 0 to 1.6 for a light aicraft for angles of attack betwen 0° and 17°
CL ≈ 0 to 1.3 for a fighter jet for angles of attack betwen 0° and 30°
A = The effective area of the wing or lifting surface (m2)
As with drag, the lift increases with the square of the velocity but decreases with altitude as the density of the air decreases.
The Lift over Drag Ratio: L / D or CL/ Cd at a given speed and angle of attack gives a measure of the Aerodynamic Efficiency of the aerofoil at that speed and angle of attack.
At very low speeds both the drag and the lift are also very small.
Because lift and drag forces are both proportional to the square of the velocity, the lift and drag at Mach 3 will be 9 times greater than the lift and drag at Mach 1.
Maximum range can be achieved by flying at very high altitudes where the density of the air is very low and the drag will consequently also be very low.
Aerodynamic flight is not possible above the atmosphere in the vacuum of space since the density of the ambient environment will be zero and so both the lift and drag will also be zero.
See more about Aerodynamic Lift, Drag, Angle of Attack and the Theory of Flight.