This point is critical

The critical point (CP,) also called the equi-time point (ETP,) is the point exactly halfway in terms of time.

Let's say you are riding your bike to your friends house 10km away and you ride at about 10km/H. Where is the CP? Commonsense tells you it is at the 5km mark, exactly halfway between your house and your friends house.

Now let's say you are riding in the opposite direction to Grandmas house which is at the top of a big hill, again 10km away. You can ride up the hill at 6km/H, and back at 14km/H. Where is the CP? Not at the 5km mark, because it takes you more time to ride up the hill than down. We need to use the speeds to find a ratio and apply to the distance, thus;

CP = D x Vhome/(Vout+Vhome)

CP = 10 x 14/(6+14)

CP = 7km

To test, we'll see how long it takes us to ride from the CP to our destination, and how long back.

3km / 6kmH = 0.5 * 60 = 30min

7km / 14kmH = 0.5 * 60 = 30min

So here we are, riding to grandmas house, when we remember we have to call mum to let her know we are on our way to grandmas house, do we turn back and let her know, of do we ride on to grandmas house and ring her from there? Up until the 3km/7km mark, if we turn around it will take less then 30min to ride home. After the 3km/7km mark, it will take less then 30min to ride on.

This is similar to the problem that aircraft face in flight when deciding to continue on or return home in response to changed circumstances. In nil wind, the CP is exactly halfway, but if this is any headwind or tailwind (most likely) the CP will move, and it is worth noting it will move into the headwind (uphill.)

We are flying a B727 from Mount Isa to Alice Springs, a distance of 1050NM. Our cruising speed is 450kts TAS, we have a 50kt headwind and we are past the halfway mark with only 500NM to go. A passenger on board has a medical emergency. 500NM ahead of us is Alice Springs hospital and 550NM behind us is Mount Isa hospital and a whole lot of nothing in between. Do we go on or turn back? Alice is closer, but due to the ground speed difference maybe it will take less time to fly back to Isa.

CP = 1050 * 500 / (400+500)

CP = 583NM from Mount Isa (remembering the CP always moves into wind.)
or
CP = 467NM from Alice Springs.

In this case we go back.

To test, let's fly on to Alice from the CP

400/467*60 = 51min
500/583*60 = 51min.

RESULT.