Into space, by the skin of their teeth
Jun. 21st, 2004 06:51 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
major_clangerSpaceflight Now has a regularly-updated news feed on the SpaceShipOne flight and subsequent press conference.
Scaled Composites are claiming that apogee was 100.124 km, so they hit their target by just 124 metres. Whilst 'watching' the flight (i.e. hitting 'refresh' on my web browser whilst pointed at the above) I did a quick back-of-the-envelope calculation to the effect that every 1 metre-per-second of speed at the end of the ascent burn added 100 metres to the peak altitude. Since SS1 was accelerating at 3g - or about thirty metres per second every second - at burnout, this means that if the burn had been, oh, 1/20 of a second shorter, it wouldn't have reached 100 km.
Now that is cutting it fine.
Some rough Rocket Science:
SS1's flight profile is to accelerate vertically to Mach 3 (about 1000 m/s) at about 50 km altitude, then coast to apogee before coming straight back down again. If you are travelling upwards at velocity v then your peak altitude will be
s = (v^2)/2a
where a is acceleration due to gravity. That's 9.81 m/s/s, or about 10 for our purposes. So, peak altitude above burnout height h will be
peak = h + (v^2)/2a
and if h = 50,000 and v = 1000, then we get a peak of 100,000 m as required.
So what is the sensitivity of this to burnout velocity? We want to know the rate of change of peak altitude with respect to v, which we get by differentiating:
d(peak)/dv = 2v/2a = v/a
When v = 1000 m/s, d(peak)/dv = 100 m per m/s (a bit of an odd unit, but it's what we're after). So a very small change in burnout speed has quite a big impact of peak altitude, and this effect is made even worse because SS1 is lightest, and thus accelerating fastest, just before burnout. So a tiny change in burn length can significantly affect how high it ends up.
MC
Scaled Composites are claiming that apogee was 100.124 km, so they hit their target by just 124 metres. Whilst 'watching' the flight (i.e. hitting 'refresh' on my web browser whilst pointed at the above) I did a quick back-of-the-envelope calculation to the effect that every 1 metre-per-second of speed at the end of the ascent burn added 100 metres to the peak altitude. Since SS1 was accelerating at 3g - or about thirty metres per second every second - at burnout, this means that if the burn had been, oh, 1/20 of a second shorter, it wouldn't have reached 100 km.
Now that is cutting it fine.
Some rough Rocket Science:
SS1's flight profile is to accelerate vertically to Mach 3 (about 1000 m/s) at about 50 km altitude, then coast to apogee before coming straight back down again. If you are travelling upwards at velocity v then your peak altitude will be
s = (v^2)/2a
where a is acceleration due to gravity. That's 9.81 m/s/s, or about 10 for our purposes. So, peak altitude above burnout height h will be
peak = h + (v^2)/2a
and if h = 50,000 and v = 1000, then we get a peak of 100,000 m as required.
So what is the sensitivity of this to burnout velocity? We want to know the rate of change of peak altitude with respect to v, which we get by differentiating:
d(peak)/dv = 2v/2a = v/a
When v = 1000 m/s, d(peak)/dv = 100 m per m/s (a bit of an odd unit, but it's what we're after). So a very small change in burnout speed has quite a big impact of peak altitude, and this effect is made even worse because SS1 is lightest, and thus accelerating fastest, just before burnout. So a tiny change in burn length can significantly affect how high it ends up.
MC
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no subject
Date: 2004-06-21 11:27 am (UTC)no subject
Date: 2004-06-21 11:49 am (UTC)no subject
Date: 2004-06-21 12:03 pm (UTC)MC
no subject
Date: 2004-06-21 01:26 pm (UTC)Methinks they'll need to get the motor and/or flight profile more finely tuned. It looks as though their design will barely meet its design parameters.
no subject
Date: 2004-06-21 02:23 pm (UTC)