1. Gravitational Constant

The gravitational constant represents the acceleration due to gravity:

2. Air Density

The density of air at sea level, which affects air resistance:

3. Drag Coefficient

The drag coefficient represents the aerodynamic properties of the rocket:

4. Launch Angle in Radians

The launch angle is converted from degrees to radians:

5. Sin(2θ)

We calculate the trigonometric factor \( \sin(2 \theta) \) to adjust for the launch angle:

6. Drag Factor Constant (k)

The drag factor combines the drag coefficient, air density, and cross-sectional area:

7. Drag-Modified Range Factor

The drag factor modifies the range based on air resistance:

8. Final Range Calculation

The final range is calculated by adjusting for the drag factor:

The blast radius is calculated using the following formula:

Where:

• R is the blast radius
• W is the explosive yield in kilograms
• Z is the scaled distance, calculated using the Kingery-Bulmash scaling law

To calculate Z (scaled distance), the formula is:

Where:

• P is the overpressure in Pascals
• Z is the scaled distance in meters

Multi-stage Propulsion

ICBMs like the RS-24 Yars use multiple stages of propulsion, with each stage propelling the missile to much higher speeds than conventional rockets. These stages help overcome Earth's gravity and reach much higher altitudes. Each stage separates after burning out, and the missile's remaining mass continues at much higher velocities than a single-stage system, drastically increasing range.

Sub-orbital Trajectory

ICBMs follow a sub-orbital flight path, meaning they exit the Earth's atmosphere for most of their flight and then re-enter during the terminal phase. In space, there is no air resistance, which allows the missile to travel much farther with minimal energy loss. The altitude of an ICBM trajectory can be hundreds of kilometers (the RS-24 Yars can reach altitudes of up to 1,000 km). This high-altitude flight allows the missile to cover much more ground than a projectile that remains within the atmosphere.

Earth’s Curvature

The Earth’s curvature plays a significant role in long-range trajectories like those of ICBMs. While shorter-range rockets follow a near-straight or parabolic path, ICBMs can benefit from the Earth’s round shape, effectively extending their reach over the surface. Our basic formula doesn't account for the fact that the Earth is curved.

Re-entry Phase

The final phase of an ICBM involves re-entry into the Earth's atmosphere, during which the warhead is guided back to its target. The missile is designed to withstand the extreme heat and speed of re-entry, which also increases its accuracy and range. At this phase, gravity and atmospheric drag are balanced by the high speed acquired during the ballistic phase.

Optimized Trajectories

Modern ICBMs use computer-guided optimized trajectories to maximize range and accuracy. These are not simple ballistic trajectories but are tailored using advanced calculations that involve gravitational slingshots, changing air densities, and precise guidance.

Why the Formula Underestimates ICBM Range:

• Higher Altitudes and Speeds: Our formula assumes the projectile remains in the Earth's atmosphere, but ICBMs exit it, reducing the effect of gravity and air resistance.
• Multi-stage Propulsion: ICBMs use multiple stages to reach speeds far beyond those of conventional rockets.
• Earth's Curvature: The Earth's curvature allows longer distances to be covered than our formula assumes.