### The Kill-zone: How to Dodge a Sniper’s Bullet

Under mild simplifying assumptions, a projectile follows a parabolic trajectory. This results from Newton’s law of motion. Thus, for a fixed energy, there is an accessible region around the firing point comprising all the points that can be reached. We will derive a mathematical description for this kill-zone (the term kill-zone, used for dramatic effect, is the region embracing all the points that can be reached by a sniper’s bullet, given a fixed muzzle velocity).

Family of trajectories with fixed initial speed and varying launch angles. Two particular trajectories are shown in black.

The Laws of Motion

We look at motion in a vertical plane, and measure distances ${x}$ horizontally and ${y}$ vertically, with the launch-point as origin. If air drag is neglected, the only force is the downward gravitation and the equations of motion (dots denoting time derivatives) are

$\displaystyle \begin{array}{rcl} m \dot u &= m \ddot x &= 0 \\ m \dot v &= m \ddot y &= -mg \end{array}$

The initial position is ${(x_0,y_0)=(0,0)}$ and the initial velocity is ${(u_0,v_0)}$. Since the acceleration of gravity ${g}$ is constant, the equations are easily integrated. The first integration gives

$\displaystyle \begin{array}{rcl} u &= \dot x &= u_0 \\ v &= \dot y &= v_0 - g t \end{array}$

Thus, the horizontal speed ${u}$ remains constant and the vertical speed ${v}$ decreases linearly with time. Another integration gives the positions:

$\displaystyle \begin{array}{rcl} x &= u_0 t \\ y &= v_0 t - \frac{1}{2} g t^2 \end{array}$

Using the first of these to replace ${t}$ by ${x/u_0}$, we get

$\displaystyle y = \left(\frac{v_0}{u_0}\right) x - \left(\frac{g}{2 u_0^2}\right) x^2 \ \ \ \ \ (1)$

This is an equation for a parabola, concave downwards with a vertical axis. It is obvious that ${y=0}$ when ${x = 0}$ and when ${x = (u_0 v_0)/g}$.

The launch angle ${\theta}$ is given in terms of the initial velocity components:

$\displaystyle \tau = \tan\theta = \frac{v_0}{u_0}$

The initial speed squared is ${V_0^2 = u_0^2 + v_0^2}$ and the components may now be written

$\displaystyle u_0 = V_0 \cos\theta \qquad\mbox{and}\qquad v_0 = V_0 \sin\theta$

Bounding the Kill-zone

Let’s consider an arbitrary target point ${(X,Y)}$ and ask if it lies on one or more of the trajectories. With initial speed ${V_0}$ and launch angle ${\theta}$, the trajectory is given by equation (1), now written

$\displaystyle y = \left(\tan\theta\right) x - \left(\frac{g}{2 V_0^2\cos^2\theta}\right) x^2 \ \ \ (2)$

Noting that ${\tau = \tan\theta}$ and ${\cos^2\theta = 1/\sec^2\theta = 1/(1+\tau^2)}$, and assuming that ${(X,Y)}$ is on the trajectory, we can write

$\displaystyle Y = \tau X - (1+\tau^2)\left(\frac{g}{2 V_0^2}\right) X^2 \ \ \ \ \ (3)$

This is a quadratic in ${\tau}$ which we write

$\displaystyle \left(\frac{g X^2}{2 V_0^2}\right) \tau^2 - X \tau + \left[ Y + \left(\frac{g X^2}{2 V_0^2}\right) \right] = 0 \ \ \ \ \ (4)$

The accessible points are those that yield real solutions for ${\tau}$. The criterion for this is the positivity of the discriminant

$\displaystyle \Delta = X^2 - 4 \left(\frac{g X^2}{2 V_0^2}\right)\left[ Y + \left(\frac{g X^2}{2 V_0^2}\right) \right]$

For ${\Delta > 0}$ there are two real roots and two trajectories passing through ${(X,Y)}$. For ${\Delta < 0}$ there are no real roots and no trajectories through ${(X,Y)}$. The accessible region is delimited by requiring ${\Delta = 0}$, which gives us

$\displaystyle Y = \left(\frac{V_0^2}{2g}\right) - \left(\frac{g}{2 V_0^2}\right) X^2 \ \ \ \ \ (5)$

This is a parabola and it is the envelope of all the parabolic trajectories. It is shown as a red curve in the Figure below.

Family of trajectories (blue) and their envelope curve (red).

A More General Approach

There is a general method of constructing the envelope of a family of curves. Suppose each curve is determined by a value of the parameter ${\theta}$, and the family can be specified by the equation ${f(x,y,\theta)=0}$. In the case considered above we have

$\displaystyle f(x,y,\theta) \equiv \left(\tan\theta\right) x - \left(\frac{g}{2 V_0^2\cos^2\theta}\right) x^2 - y \ \ \ \ \ (6)$

The envelope is determined by the simultaneous equations

$\displaystyle f(x,y,\theta) = 0 \qquad\mbox{and}\qquad \frac{\partial f}{\partial\theta} = 0$

Eliminating the parameter ${\theta}$ between these, we get an equation of the form ${g(x,y) = 0}$ for the envelope. The method gives the same result (5) as before.

Final Remark

We have seen that a sniper can shoot anyone who is within the envelope parabola. However, several practical issues have been ignored. In particular, we note that, since the energy of the projectile is conserved and its potential energy increases linearly with height, the kinetic energy must decrease linearly. Thus, close to the envelope apex the speed of the bullet may be insufficient to inflict much damage.

Acknowledgement: The problem of the envelope of parabolic trajectories was suggested by Dr Mark Dukes of UCD School of Mathematics and Statistics.