Question

For anti-ballistic missile system, the time of flight tf is determined by the initial speed v0 of the missile and the maximum range r of the incoming missile. find their relationship.

Answer 1

Refer to the diagram shown below.

The following discussion assumes a simplistic analysis that ignores air resistance and variations in the terrain that the missile travels over.

Let the launch velocity be V₀ at an angle of θ relative to the horizontal.
The horizontal component of velocity is V₀ cosθ.
If the time of flight is $t_{f}$, then
$r=V_{o} \, t_{f}$
where r = the range of the missile.

Also, the time, t, when the missile is at ground level is given by
$0=V_{o} sin\theta \, t- \frac{1}{2}gt^{2}$
where g = acceleration due to gravity.

t = 0 corresponds to when the missile is launched. Therefore
$t_{f} = \frac{2V_{o}sin\theta}{g}$

Therefore
$r= \frac{2V_{o}^{2} sin\theta cos\theta}{g} = \frac{V_{o}^{2} sin(2\theta)}{g}$

Typically, θ=45° to achieve maximum range, so that
$r= \frac{V_{o}^{2}}{g}$

This analysis is more applicable to a scud missile rather than a powered, guided missile.

Answer:
$t_{f} = \frac{r}{V_{o} cos\theta} \\\\ r= \frac{V_{o}^{2} sin(2\theta)}{g}$
Usually, θ=45°