It can't exceed its max velocity and shooting at an almost vertical apogee will limit even further rentry speed. Is it Prithvi I being fired because that will limit even further.
That is Mach 10-12. Can a Prithvi I get that fast much less do it on a near verticle apogee? I hear it is only Mach 5.
Black Sparrow simulates Scud B at 1.4km/s rentry speed, the terminal speed of Prithvi is 1.7km/s and will go slower making a near verticle apogee at booster burnout. Just because it falls doesn't mean it will go faster... it slows down.
You are looking at Prithvi's operational capabilities. The velocities achieved by a warhead has everything to do with gravity and not propulsion. The higher the warhead separation, the greater is the reentry velocity.
Black Sparrows have simulated from Scuds to Shabab-3. To simulate Shabab-3, the F-15 flew to a height of 27Km before firing the missile, pretty impressive.
Here 27Km is not an operational capability of the F-15, regardless it was achieved for tests. Prithvi and any other ballistic missile can do the same by crossing the threshold of what is perceived as normal for the system. The Prithvi being used for BMD tests don't have to be operational missiles, they would have obviously been modified specifically for the tests in order to simulate larger missiles.
As for the math involved,
v[SUP]2[/SUP] = u[SUP]2[/SUP] + 2as
where v = final velocity
u = initial velocity
a = acc due to gravity
s = distance.
At heights of 200Km and considering initial velocity is zero, the velocity achieved is close to 2Km/s. Now this is only considering the warhead was merely dropped. If there was initial velocity then the speed will be greater. Add an equivalent initial speed and you have a simulated 2000Km IRBM. Air resistance plays foul too, so that needs to be catered for. I think the tests were used to simulate a 2000Km IRBM. Agni II has an apogee of 405Km and a speed of 4Km/s on reentry.
In case of Black Sparrow we know a high initial velocity was achieved after it was fired.
Btw, falling down against Earth's gravity increases speed, it does not slow down.
The success of the first phase BMD highly depends on howmany batteries the IAF purchases. If the IAF buys a lot then we know the system is good enough for what we need. The problem arises only if IAF rejects the BMD.