Are you unfamiliar with the concept of expressing average velocity over time?

Great - now we're getting somewhere, as you are at last explaining how you're getting to your erroneous statements. Average velocity would only work if the rocket's acceleration was constant. But it is far from constant. At launch, there is only a small amount of excess thrust over the weight of the rocket. But close to H_{bo}, the rocket is a lot lighter and yet still has the same thrust. So most of the accelerating happens in the latter stages of the burn. The Hwasong 15, for example, was estimated to have reached 7.17km/s at the end of its burn. At that velocity, it would take around 12 minutes to decelerate to 0 at 9.81ms^{-2}

At which point you'll probably say 'so how come it flew for 50 minutes? Surely 12 x 2 = 24?'

Which then leads us to the other complicating factor that you aren't considering, which is the progressive reduction in g as you get further away from earth. At 4000km, for example, the ICBM would only experience a g of 3.7ms^{-2}. That's why the calculations get complex very quickly - you've got variable mass and g, and a rotating planet.

Of course, I guess you disagree with the rotating round planet bit, and probably the reducing g bit as well. That's fine...but the burden then falls on you to explain what exactly did happen to the rocket if it flew for 50 odd minutes and only went 950km from the launch site, and how far it would be able to travel if it was launched at a shallower angle.

I completely understand what you wrote.

I also completely understand that a rocket will completely accelerate until the end of its burn, at which point it will cease to continue the process of acceleration.

However, we are still at

*t**+5*, and we are still left with 250km altitude. and we are still left with the average rate of acceleration at 3,000 km/h.

So, splitting down to minutes,

*t**+1*,

*t**+2*,

*etc*, given, for instance, the missile could obtain 1,000 km/h at

*t**+1*, 2,000 km/h at

*t**+2*,

**etc.**, up to 5,000 km/h, in order to average 3,000 km/h. Other values are possible over the span of 5 minutes, but the

**sum of these values**/

*t* cannot exceed 3,000 km/h.

So, how does a missile without engine burn, as you stated, accelerate to gain 4250 km in altitude within 48 minutes, fighting against

*g*?