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).

**The Laws of Motion**

We look at motion in a vertical plane, and measure distances horizontally and 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

The initial position is and the initial velocity is . Since the acceleration of gravity is constant, the equations are easily integrated. The first integration gives

Thus, the horizontal speed remains constant and the vertical speed decreases linearly with time. Another integration gives the positions:

Using the first of these to replace by , we get

This is an equation for a parabola, concave downwards with a vertical axis. It is obvious that when and when .

The launch angle is given in terms of the initial velocity components:

The initial speed squared is and the components may now be written

**Bounding the Kill-zone**

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

Noting that and , and assuming that is on the trajectory, we can write

This is a quadratic in which we write

The accessible points are those that yield real solutions for . The criterion for this is the positivity of the discriminant

For there are two real roots and two trajectories passing through . For there are no real roots and no trajectories through . The accessible region is delimited by requiring , which gives us

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.

**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 , and the family can be specified by the equation . In the case considered above we have

The envelope is determined by the simultaneous equations

Eliminating the parameter between these, we get an equation of the form 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.