General
Knowing the time-to-go ($t_{go}$) is essential for advanced missile systems, simply because it defines which phase of the intercept the missile is in. This helps to inform logic and guidance decisions as well as how aggressive they are, which in turn shall maximize the probability of intercept compared to simpler algorithms such as proportional guidance.
Application examples
Such logic decisions may be for example when to switch on the missiles active radar system, which in turn might alert your enemy of your presence, which motivates to do that as late as possible. The AMRAAM missile uses such a system. Other missiles change their flight behaviour for example the MBDA Meteor throttles up and down to optimize its energy when reaching its target and even changes its control strategy (bank-to-turn for skid-to-turn) when it is about to hit its target even though that will starve its ramjet engine. Another example is that of a so-called "loft" whereby long-range missiles loft their trajectory to fly through upper parts of the atmosphere in order to take advantage of lower air-drag. Again this kind of behaviour cannot be realized without the compution of your targeted impact point and therefore an explicit knowledge of $t_{go}$.
Comparison to proportional guidance
Concerning your doubts about "practicality" of such systems:
- Keep in mind that these are computer-controlled missiles with (in case of modern missiles) quite a lot of computing power. It is no different to the missile if it has to perform such calculations during its flight or in the endgame as long as there is enough computing power available (which of course was validated by the designers beforehand).
- You imply that the proportional navigation would be the better solution which would negate the need for the computation of $t_{go}$. As given by the examples above, $t_{go}$ can be used to increase the chance of hitting a target. Additionally proportional navigation is not perfect either. For example, if you employ PN against a target very far away which only maneuvers slightly (for example by executing a very minor sinusoid motion) your target point will move very dramatically. This results in great maneuvering on the side of the interceptor missile as you try to keep up with this target point. This excessive maneuvers leads to the missile bleeding off a lot of its energy. Allegedly, this happend with Patriot missile defending against Iraqi modified scud missiles which became unstable in the air, however I cannot find the source in the moment. The estimation of $t_{go}$ could inform a decision to adapt your guidance behaviour in order to rule out premature energy loss in midcourse flight.
How to compute $t_{go}$
It is as simple as y dividing your velocity by remaining distance $t_{go} = \frac{x_{go}}{V_{missile}}$. The trick is to know these to parameters over your flight. Modern missile such as the AMRAAM explicitly calculate their trajectory over their flight path until the predicted point of intercept and therefore know exactly how far they intend to fly. From this they can then explicitly calculate how long they will fly. This is computationally expensive, however as stated before modern missile do contain the necessary computing power to carry out these kinds of calculations and they use it! For example look at the different flight profiles of a Javelin missile or read around a bit on the different modes and functionalities of modern missiles, you will be amazed of what kind of complex designs and logic trees these include.
