Recent Claim by Musk concerning hot 2nd stage start

[bknight]

In a recent interview Musk has stated that a hot start of the second stage while still attached to the first stage will "conservatively result in 10% more payload into LEO.  Is this correct?
https://twitter.com/i/spaces/1OyJAVEdkMaxb
Note this is a 49 minute interview and there were sound issues for the first 16:30 .  Skip ahead, if you are listening.
Now I realize that shutting down the first stage and letting the vehicles separate prior to second stage results in the second stage decelerating until it begins to accelerated again.   So hot starting will result in more payload to LEO, but 10%?  SpaceX is installing an extension to the inner stage and beefing up the first stage to prevent damage to the first stage.

[smartcooky]

Not sure of the engineering calculations, but it sound reasonable. The payload only comprises a relatively small percentage of the second stage mass. Here are some figures for Falcon 9, non-expended

Empty Mass:...  3900 kg
Propellant:.... 92,670 kg
Payload:....... 22,800 kg
--------
Total.... ..... 119370 kg

The payload is 19% of the total mass of the second stage at firing

Increase the mass of the payload by 10% = 25080 Kg

Empty Mass:...  3900 kg
Propellant:.... 92,670 kg
Payload:....... 25,080 kg
--------
Total.... ..... 121650 kg

The payload is now 21% of the second stage at firing

So in reality, there is only a 2% increase in mass to orbit

(provided my back-of-the-envelope math is correct)

ETA: Of course, this is very simplified. I have take no account of the reduction in overall mass as the propellant is burned, but it doesn't seem unreasonable to me that firing the second stage engine a few seconds early, and before it has a chance to start to decelerate might allow a couple of additional % points of mass to LEO.

[bknight]

Well I created a velocity spreadsheet for A11 with the help of BobB a few years ago and the numbers seem more like your 2% that 10%.  And remember he said conservative 10%.  I posted part of the spreadsheet in CQ, and I have shut it down currently and will be back tomorrow and post it here.

[smartcooky]

Fair enough, and if Musk is making exaggerated claims, well, it wouldn't be the first time, would it?

I'm just pointing out that a 10% increase in payload is a significantly less than that in overall second stage mass. Remember that the second stage goes all the way to orbit with the payload, and in the case of Falcon 9 and Falcon Heavy, carries some extra fuel for a de-orbit burn.

Also, any mass you can strip from the second stage can go to payload. For example, when Rocketlab's Electron made its first flight six years ago, it had a maximum payload to LEO of 225kg. This has gone up to 300 KG, because in the intervening six years, the lithium polymer batteries they use to drive the fuel pumps have gotten smaller and lighter while still delivering the same capacity.

In the case of Falcon 9, with a "hot start" on the second stage, the fact that it will be traveling at a higher velocity might mean it needs to carry less fuel... which equals less mass. I just had a quick look at a Starlink launch - at MECO (2m33s) the whole stack is traveling at 8184 km/h, and drops to 8057 km/h at the moment of stage separation. When SES happens, the second stage does not get back up to 8187 km/h until 2m 49s.     
 

PS: Its worth noting that the Russians have used hot staging on their rockets since the late 1050s. India and China also use it.

[bknight]

Fair enough, and if Musk is making exaggerated claims, well, it wouldn't be the first time, would it?

I'm just pointing out that a 10% increase in payload is a significantly less than that in overall second stage mass. Remember that the second stage goes all the way to orbit with the payload, and in the case of Falcon 9 and Falcon Heavy, carries some extra fuel for a de-orbit burn.

Also, any mass you can strip from the second stage can go to payload. For example, when Rocketlab's Electron made its first flight six years ago, it had a maximum payload to LEO of 225kg. This has gone up to 300 KG, because in the intervening six years, the lithium polymer batteries they use to drive the fuel pumps have gotten smaller and lighter while still delivering the same capacity.

In the case of Falcon 9, with a "hot start" on the second stage, the fact that it will be traveling at a higher velocity might mean it needs to carry less fuel... which equals less mass. I just had a quick look at a Starlink launch - at MECO (2m33s) the whole stack is traveling at 8184 km/h, and drops to 8057 km/h at the moment of stage separation. When SES happens, the second stage does not get back up to 8187 km/h until 2m 49s.     
 

PS: Its worth noting that the Russians have used hot staging on their rockets since the late 1050s. India and China also use it.

That observation (16 sec) seems a bit slow, but your eye not mine, just saying.
   Vehicle Mass                        Acceleration            Space-fixed velocity         
Time   Dry   Fuel   Total   Flow   Thrust   ISP   Throttle   Pitch   g   Horz.   Vert.   Load   Horz.   Vert.   Total   f
(s)   (kg)   (kg)   (kg)   (kg/s)   (N)   (s)   (%)   (deg)   (m/s2)   (m/s2)   (m/s2)   (g)   (m/s)   (m/s)   (m/s)   (deg)
161.63   242186   577266   819452   10716.83   27808451   304.0   100%   70.14   9.650   31.632   2.716   3.46   2339.6   673.1   2434.5   16.05
162.00   242186   575284   817470   0.00   0   0.0   0%   70.25   9.650   -0.285   -8.807   0.00   2345.4   671.9   2439.7   15.99
162.11   242186   575284   817470   0.00   0   0.0   0%   70.28   9.649   -0.284   -8.807   0.00   2345.3   671.0   2439.4   15.97
162.30   242153   575284   817470   0.00   0   0.0   0%   70.34   9.649   -0.283   -8.807   0.00   2345.3   669.3   2438.9   15.93
163.00   109139   550330   817470   0.00   0   0.0   0%   70.55   9.648   -0.268   -8.801   0.00   2345.1   663.1   2437.1   15.79
164.00   108963   550330   816857   1225.45   5104333   427.7   100%   70.84   9.646   5.639   -6.746   0.63   2347.8   655.4   2437.5   15.60
165.00   108786   549105   815631   1225.45   5104333   427.7   100%   71.14   9.644   5.662   -6.767   0.64   2353.4   648.6   2441.2   15.41
165.61   108678   548357   814884   1225.45   5104333   427.7   100%   70.82   9.643   5.658   -6.727   0.64   2356.9   644.5   2443.4   15.29
166.00   108678   547879   814406   1225.45   5104333   427.7   100%   70.61   9.642   5.656   -6.702   0.64   2359.1   641.9   2444.9   15.22
167.00   108678   546654   813181   1225.45   5104333   427.7   100%   70.08   9.640   5.649   -6.638   0.64   2364.7   635.2   2448.6   15.04
168.00   108678   545428   811955   1225.45   5104333   427.7   100%   69.55   9.638   5.641   -6.574   0.64   2370.4   628.6   2452.3   14.85
169.00   108678   544203   810730   1225.45   5104333   427.7   100%   69.03   9.636   5.632   -6.510   0.64   2376.0   622.1   2456.1   14.67
170.00   108678   542977   809504   1225.45   5104333   427.7   100%   68.50   9.634   5.623   -6.446   0.64   2381.7   615.6   2459.9   14.49
171.00   108678   541752   808279   1225.45   5104333   427.7   100%   67.97   9.632   5.613   -6.382   0.64   2387.3   609.2   2463.8   14.31
172.00   108678   540526   807053   1225.45   5104333   427.7   100%   67.44   9.631   5.603   -6.318   0.64   2392.9   602.8   2467.6   14.14
173.00   108678   539301   805828   1225.45   5104333   427.7   100%   66.91   9.629   5.592   -6.255   0.64   2398.5   596.5   2471.5   13.97
174.00   108678   538076   804602   1225.45   5104333   427.7   100%   66.38   9.627   5.580   -6.191   0.65   2404.1   590.3   2475.5   13.80
175.00   108678   536850   803377   1225.45   5104333   427.7   100%   65.86   9.625   5.568   -6.127   0.65   2409.6   584.1   2479.4   13.63
176.00   108678   535625   802152   1225.45   5104333   427.7   100%   65.33   9.624   5.555   -6.064   0.65   2415.2   578.0   2483.4   13.46
177.00   108678   534399   800926   1225.45   5104333   427.7   100%   64.80   9.622   5.541   -6.001   0.65   2420.7   572.0   2487.4   13.29
178.00   108678   533174   799701   1225.45   5104333   427.7   100%   64.27   9.620   5.527   -5.937   0.65   2426.3   566.0   2491.4   13.13
179.00   108678   531948   798475   1225.45   5104333   427.7   100%   63.74   9.618   5.512   -5.874   0.65   2431.8   560.1   2495.5   12.97
180.00   108678   530723   797250   1225.45   5104333   427.7   100%   63.21   9.617   5.497   -5.811   0.65   2437.3   554.3   2499.5   12.81
181.00   108678   529497   796024   1225.45   5104333   427.7   100%   62.68   9.615   5.481   -5.748   0.65   2442.8   548.5   2503.6   12.66
182.00   108678   528272   794799   1225.45   5104333   427.7   100%   62.16   9.614   5.465   -5.685   0.65   2448.3   542.8   2507.7   12.50
183.00   108678   527046   793573   1225.45   5104333   427.7   100%   61.63   9.612   5.448   -5.622   0.66   2453.7   537.1   2511.8   12.35
184.00   108678   525821   792348   1225.45   5104333   427.7   100%   61.10   9.610   5.430   -5.560   0.66   2459.2   531.6   2516.0   12.20
185.00   108678   524596   791123   1225.45   5104333   427.7   100%   60.57   9.609   5.412   -5.497   0.66   2464.6   526.0   2520.1   12.05
186.00   108678   523370   789897   1225.45   5104333   427.7   100%   60.60   9.607   5.425   -5.489   0.66   2470.0   520.5   2524.3   11.90
187.00   108678   522145   788672   1225.45   5104333   427.7   100%   60.63   9.606   5.437   -5.482   0.66   2475.4   515.0   2528.4   11.75
188.00   108678   520919   787446   1225.45   5104333   427.7   100%   60.66   9.604   5.450   -5.474   0.66   2480.9   509.6   2532.7   11.61
189.00   108678   519694   786221   1225.45   5104333   427.7   100%   60.69   9.603   5.462   -5.466   0.66   2486.3   504.1   2536.9   11.46
190.00   108678   518468   784995   1225.45   5104333   427.7   100%   60.73   9.601   5.475   -5.459   0.66   2491.8   498.6   2541.2   11.32
191.00   108678   517243   783770   1225.45   5104333   427.7   100%   60.76   9.600   5.487   -5.451   0.66   2497.3   493.2   2545.5   11.17
192.00   108678   516017   782544   1225.45   5104333   427.7   100%   60.79   9.598   5.500   -5.444   0.66   2502.8   487.7   2549.9   11.03
192.30   108678   515650   782177   1225.45   5104333   427.7   100%   60.80   9.598   5.504   -5.441   0.67   2504.4   486.1   2551.2   10.98


Pasting spreadsheets to forums is not good.  Anyway this in A11 and the velocities are space fixed.
For reference:
161.63 OESC S1
162.3 Stage separation.
164 ESC S2.
You can see the acceleration drops from 2.76 m/s at OESC to Earth's gravity.  That continues for ~ 2 sec. Then S2 ESC.  In this scenario the total velocity meets/exceeds first stage total velocity in ~3.5 sec. 161.63-165.  Note most of the acceleration is going into horizontal velocity also Earth's gravity effect is diminishing.  I realize that the trajectories will be different so a one to one comparison is not valid.
Note that the speadsheet averages every one second(or one row) but it is a close approximation.  BobB's sheet and mine are very similar, he helped a lot and I give him credit for sticking with me during the development.
Note also the total velocity never decreases, just slows from the vertical gravity effect, horizontal was only slowed by the atmosphere.

[smartcooky]

Fair enough, and if Musk is making exaggerated claims, well, it wouldn't be the first time, would it?

I'm just pointing out that a 10% increase in payload is a significantly less than that in overall second stage mass. Remember that the second stage goes all the way to orbit with the payload, and in the case of Falcon 9 and Falcon Heavy, carries some extra fuel for a de-orbit burn.

Also, any mass you can strip from the second stage can go to payload. For example, when Rocketlab's Electron made its first flight six years ago, it had a maximum payload to LEO of 225kg. This has gone up to 300 KG, because in the intervening six years, the lithium polymer batteries they use to drive the fuel pumps have gotten smaller and lighter while still delivering the same capacity.

In the case of Falcon 9, with a "hot start" on the second stage, the fact that it will be traveling at a higher velocity might mean it needs to carry less fuel... which equals less mass. I just had a quick look at a Starlink launch - at MECO (2m33s) the whole stack is traveling at 8184 km/h, and drops to 8057 km/h at the moment of stage separation. When SES happens, the second stage does not get back up to 8187 km/h until 2m 49s.     
 

PS: Its worth noting that the Russians have used hot staging on their rockets since the late 1050s. India and China also use it.

That observation (16 sec) seems a bit slow, but your eye not mine, just saying.
   Vehicle Mass                        Acceleration            Space-fixed velocity         
Time   Dry   Fuel   Total   Flow   Thrust   ISP   Throttle   Pitch   g   Horz.   Vert.   Load   Horz.   Vert.   Total   f
(s)   (kg)   (kg)   (kg)   (kg/s)   (N)   (s)   (%)   (deg)   (m/s2)   (m/s2)   (m/s2)   (g)   (m/s)   (m/s)   (m/s)   (deg)
161.63   242186   577266   819452   10716.83   27808451   304.0   100%   70.14   9.650   31.632   2.716   3.46   2339.6   673.1   2434.5   16.05
162.00   242186   575284   817470   0.00   0   0.0   0%   70.25   9.650   -0.285   -8.807   0.00   2345.4   671.9   2439.7   15.99
162.11   242186   575284   817470   0.00   0   0.0   0%   70.28   9.649   -0.284   -8.807   0.00   2345.3   671.0   2439.4   15.97
162.30   242153   575284   817470   0.00   0   0.0   0%   70.34   9.649   -0.283   -8.807   0.00   2345.3   669.3   2438.9   15.93
163.00   109139   550330   817470   0.00   0   0.0   0%   70.55   9.648   -0.268   -8.801   0.00   2345.1   663.1   2437.1   15.79
164.00   108963   550330   816857   1225.45   5104333   427.7   100%   70.84   9.646   5.639   -6.746   0.63   2347.8   655.4   2437.5   15.60
165.00   108786   549105   815631   1225.45   5104333   427.7   100%   71.14   9.644   5.662   -6.767   0.64   2353.4   648.6   2441.2   15.41
165.61   108678   548357   814884   1225.45   5104333   427.7   100%   70.82   9.643   5.658   -6.727   0.64   2356.9   644.5   2443.4   15.29
166.00   108678   547879   814406   1225.45   5104333   427.7   100%   70.61   9.642   5.656   -6.702   0.64   2359.1   641.9   2444.9   15.22
167.00   108678   546654   813181   1225.45   5104333   427.7   100%   70.08   9.640   5.649   -6.638   0.64   2364.7   635.2   2448.6   15.04
168.00   108678   545428   811955   1225.45   5104333   427.7   100%   69.55   9.638   5.641   -6.574   0.64   2370.4   628.6   2452.3   14.85
169.00   108678   544203   810730   1225.45   5104333   427.7   100%   69.03   9.636   5.632   -6.510   0.64   2376.0   622.1   2456.1   14.67
170.00   108678   542977   809504   1225.45   5104333   427.7   100%   68.50   9.634   5.623   -6.446   0.64   2381.7   615.6   2459.9   14.49
171.00   108678   541752   808279   1225.45   5104333   427.7   100%   67.97   9.632   5.613   -6.382   0.64   2387.3   609.2   2463.8   14.31
172.00   108678   540526   807053   1225.45   5104333   427.7   100%   67.44   9.631   5.603   -6.318   0.64   2392.9   602.8   2467.6   14.14
173.00   108678   539301   805828   1225.45   5104333   427.7   100%   66.91   9.629   5.592   -6.255   0.64   2398.5   596.5   2471.5   13.97
174.00   108678   538076   804602   1225.45   5104333   427.7   100%   66.38   9.627   5.580   -6.191   0.65   2404.1   590.3   2475.5   13.80
175.00   108678   536850   803377   1225.45   5104333   427.7   100%   65.86   9.625   5.568   -6.127   0.65   2409.6   584.1   2479.4   13.63
176.00   108678   535625   802152   1225.45   5104333   427.7   100%   65.33   9.624   5.555   -6.064   0.65   2415.2   578.0   2483.4   13.46
177.00   108678   534399   800926   1225.45   5104333   427.7   100%   64.80   9.622   5.541   -6.001   0.65   2420.7   572.0   2487.4   13.29
178.00   108678   533174   799701   1225.45   5104333   427.7   100%   64.27   9.620   5.527   -5.937   0.65   2426.3   566.0   2491.4   13.13
179.00   108678   531948   798475   1225.45   5104333   427.7   100%   63.74   9.618   5.512   -5.874   0.65   2431.8   560.1   2495.5   12.97
180.00   108678   530723   797250   1225.45   5104333   427.7   100%   63.21   9.617   5.497   -5.811   0.65   2437.3   554.3   2499.5   12.81
181.00   108678   529497   796024   1225.45   5104333   427.7   100%   62.68   9.615   5.481   -5.748   0.65   2442.8   548.5   2503.6   12.66
182.00   108678   528272   794799   1225.45   5104333   427.7   100%   62.16   9.614   5.465   -5.685   0.65   2448.3   542.8   2507.7   12.50
183.00   108678   527046   793573   1225.45   5104333   427.7   100%   61.63   9.612   5.448   -5.622   0.66   2453.7   537.1   2511.8   12.35
184.00   108678   525821   792348   1225.45   5104333   427.7   100%   61.10   9.610   5.430   -5.560   0.66   2459.2   531.6   2516.0   12.20
185.00   108678   524596   791123   1225.45   5104333   427.7   100%   60.57   9.609   5.412   -5.497   0.66   2464.6   526.0   2520.1   12.05
186.00   108678   523370   789897   1225.45   5104333   427.7   100%   60.60   9.607   5.425   -5.489   0.66   2470.0   520.5   2524.3   11.90
187.00   108678   522145   788672   1225.45   5104333   427.7   100%   60.63   9.606   5.437   -5.482   0.66   2475.4   515.0   2528.4   11.75
188.00   108678   520919   787446   1225.45   5104333   427.7   100%   60.66   9.604   5.450   -5.474   0.66   2480.9   509.6   2532.7   11.61
189.00   108678   519694   786221   1225.45   5104333   427.7   100%   60.69   9.603   5.462   -5.466   0.66   2486.3   504.1   2536.9   11.46
190.00   108678   518468   784995   1225.45   5104333   427.7   100%   60.73   9.601   5.475   -5.459   0.66   2491.8   498.6   2541.2   11.32
191.00   108678   517243   783770   1225.45   5104333   427.7   100%   60.76   9.600   5.487   -5.451   0.66   2497.3   493.2   2545.5   11.17
192.00   108678   516017   782544   1225.45   5104333   427.7   100%   60.79   9.598   5.500   -5.444   0.66   2502.8   487.7   2549.9   11.03
192.30   108678   515650   782177   1225.45   5104333   427.7   100%   60.80   9.598   5.504   -5.441   0.67   2504.4   486.1   2551.2   10.98


Pasting spreadsheets to forums is not good.  Anyway this in A11 and the velocities are space fixed.
For reference:
161.63 OESC S1
162.3 Stage separation.
164 ESC S2.
You can see the acceleration drops from 2.76 m/s at OESC to Earth's gravity.  That continues for ~ 2 sec. Then S2 ESC.  In this scenario the total velocity meets/exceeds first stage total velocity in ~3.5 sec. 161.63-165.  Note most of the acceleration is going into horizontal velocity also Earth's gravity effect is diminishing.  I realize that the trajectories will be different so a one to one comparison is not valid.
Note that the speadsheet averages every one second(or one row) but it is a close approximation.  BobB's sheet and mine are very similar, he helped a lot and I give him credit for sticking with me during the development.
Note also the total velocity never decreases, just slows from the vertical gravity effect, horizontal was only slowed by the atmosphere.

That is  very interesting Mr. Knight. Looks like a lot of work went into that.

IIRC, Saturn V staging involved retro-rockets (eight IIRC, fitted at the base of each of the outer F-1 engines - two each engine). They were fired for about a second, and actually decelerated the first stage during staging. Same again for second stage staging using four solid-fueled retro-rockets located in a conical adapter at the front end of the second stage.  They did this because they were worried about the discarded stages running into the back of the next stages as they slowed down after engine cut-off. Clearly, stages slowing on MECO and SECO was an issue that needed to be dealt with. The Russians dealt with it by using open frameworks to connect stages and hot starting the stages.

The question for me is, does that slowing down impact the dynamics sufficiently to affect payload in any significant way?

[Zakalwe]

I am not an engineer and I do not have the ability to run the numbers on this.

Hot staging means that the number of seconds where the stack is in a ballistic curve is reduced. Which means that you are not losing the momentum that the first stage has built up up to the point of MECO prior to normal staging. As soon as those engines cut off gravity will start to do it's work and reduce the momentum that the booster has achieved.  This is work that has to be recovered by the second stage. Bear in mind that the second stage has to recover this work using Vacuum Raptors which are then operating in atmosphere, so not at their peak efficiency. I think that this might be the key to it.


Of course, there may be a weight penalty by having to add extra shielding to the booster? And you are burning the booster a little bit longer will mean that more fuel is consumed

[Zakalwe]

Another thought....how does Starship manage ullage prior to vacuum Raptor ignition? Hot staging removes the need for any ullage motors (or firing of the reaction controls for ullage purposes), so their is a weight saving there. 

[bknight]

Fair enough, and if Musk is making exaggerated claims, well, it wouldn't be the first time, would it?

I'm just pointing out that a 10% increase in payload is a significantly less than that in overall second stage mass. Remember that the second stage goes all the way to orbit with the payload, and in the case of Falcon 9 and Falcon Heavy, carries some extra fuel for a de-orbit burn.

Also, any mass you can strip from the second stage can go to payload. For example, when Rocketlab's Electron made its first flight six years ago, it had a maximum payload to LEO of 225kg. This has gone up to 300 KG, because in the intervening six years, the lithium polymer batteries they use to drive the fuel pumps have gotten smaller and lighter while still delivering the same capacity.

In the case of Falcon 9, with a "hot start" on the second stage, the fact that it will be traveling at a higher velocity might mean it needs to carry less fuel... which equals less mass. I just had a quick look at a Starlink launch - at MECO (2m33s) the whole stack is traveling at 8184 km/h, and drops to 8057 km/h at the moment of stage separation. When SES happens, the second stage does not get back up to 8187 km/h until 2m 49s.     
 

PS: Its worth noting that the Russians have used hot staging on their rockets since the late 1050s. India and China also use it.

That observation (16 sec) seems a bit slow, but your eye not mine, just saying.
   Vehicle Mass                        Acceleration            Space-fixed velocity         
Time   Dry   Fuel   Total   Flow   Thrust   ISP   Throttle   Pitch   g   Horz.   Vert.   Load   Horz.   Vert.   Total   f
(s)   (kg)   (kg)   (kg)   (kg/s)   (N)   (s)   (%)   (deg)   (m/s2)   (m/s2)   (m/s2)   (g)   (m/s)   (m/s)   (m/s)   (deg)
161.63   242186   577266   819452   10716.83   27808451   304.0   100%   70.14   9.650   31.632   2.716   3.46   2339.6   673.1   2434.5   16.05
162.00   242186   575284   817470   0.00   0   0.0   0%   70.25   9.650   -0.285   -8.807   0.00   2345.4   671.9   2439.7   15.99
162.11   242186   575284   817470   0.00   0   0.0   0%   70.28   9.649   -0.284   -8.807   0.00   2345.3   671.0   2439.4   15.97
162.30   242153   575284   817470   0.00   0   0.0   0%   70.34   9.649   -0.283   -8.807   0.00   2345.3   669.3   2438.9   15.93
163.00   109139   550330   817470   0.00   0   0.0   0%   70.55   9.648   -0.268   -8.801   0.00   2345.1   663.1   2437.1   15.79
164.00   108963   550330   816857   1225.45   5104333   427.7   100%   70.84   9.646   5.639   -6.746   0.63   2347.8   655.4   2437.5   15.60
165.00   108786   549105   815631   1225.45   5104333   427.7   100%   71.14   9.644   5.662   -6.767   0.64   2353.4   648.6   2441.2   15.41
165.61   108678   548357   814884   1225.45   5104333   427.7   100%   70.82   9.643   5.658   -6.727   0.64   2356.9   644.5   2443.4   15.29
166.00   108678   547879   814406   1225.45   5104333   427.7   100%   70.61   9.642   5.656   -6.702   0.64   2359.1   641.9   2444.9   15.22
167.00   108678   546654   813181   1225.45   5104333   427.7   100%   70.08   9.640   5.649   -6.638   0.64   2364.7   635.2   2448.6   15.04
168.00   108678   545428   811955   1225.45   5104333   427.7   100%   69.55   9.638   5.641   -6.574   0.64   2370.4   628.6   2452.3   14.85
169.00   108678   544203   810730   1225.45   5104333   427.7   100%   69.03   9.636   5.632   -6.510   0.64   2376.0   622.1   2456.1   14.67
170.00   108678   542977   809504   1225.45   5104333   427.7   100%   68.50   9.634   5.623   -6.446   0.64   2381.7   615.6   2459.9   14.49
171.00   108678   541752   808279   1225.45   5104333   427.7   100%   67.97   9.632   5.613   -6.382   0.64   2387.3   609.2   2463.8   14.31
172.00   108678   540526   807053   1225.45   5104333   427.7   100%   67.44   9.631   5.603   -6.318   0.64   2392.9   602.8   2467.6   14.14
173.00   108678   539301   805828   1225.45   5104333   427.7   100%   66.91   9.629   5.592   -6.255   0.64   2398.5   596.5   2471.5   13.97
174.00   108678   538076   804602   1225.45   5104333   427.7   100%   66.38   9.627   5.580   -6.191   0.65   2404.1   590.3   2475.5   13.80
175.00   108678   536850   803377   1225.45   5104333   427.7   100%   65.86   9.625   5.568   -6.127   0.65   2409.6   584.1   2479.4   13.63
176.00   108678   535625   802152   1225.45   5104333   427.7   100%   65.33   9.624   5.555   -6.064   0.65   2415.2   578.0   2483.4   13.46
177.00   108678   534399   800926   1225.45   5104333   427.7   100%   64.80   9.622   5.541   -6.001   0.65   2420.7   572.0   2487.4   13.29
178.00   108678   533174   799701   1225.45   5104333   427.7   100%   64.27   9.620   5.527   -5.937   0.65   2426.3   566.0   2491.4   13.13
179.00   108678   531948   798475   1225.45   5104333   427.7   100%   63.74   9.618   5.512   -5.874   0.65   2431.8   560.1   2495.5   12.97
180.00   108678   530723   797250   1225.45   5104333   427.7   100%   63.21   9.617   5.497   -5.811   0.65   2437.3   554.3   2499.5   12.81
181.00   108678   529497   796024   1225.45   5104333   427.7   100%   62.68   9.615   5.481   -5.748   0.65   2442.8   548.5   2503.6   12.66
182.00   108678   528272   794799   1225.45   5104333   427.7   100%   62.16   9.614   5.465   -5.685   0.65   2448.3   542.8   2507.7   12.50
183.00   108678   527046   793573   1225.45   5104333   427.7   100%   61.63   9.612   5.448   -5.622   0.66   2453.7   537.1   2511.8   12.35
184.00   108678   525821   792348   1225.45   5104333   427.7   100%   61.10   9.610   5.430   -5.560   0.66   2459.2   531.6   2516.0   12.20
185.00   108678   524596   791123   1225.45   5104333   427.7   100%   60.57   9.609   5.412   -5.497   0.66   2464.6   526.0   2520.1   12.05
186.00   108678   523370   789897   1225.45   5104333   427.7   100%   60.60   9.607   5.425   -5.489   0.66   2470.0   520.5   2524.3   11.90
187.00   108678   522145   788672   1225.45   5104333   427.7   100%   60.63   9.606   5.437   -5.482   0.66   2475.4   515.0   2528.4   11.75
188.00   108678   520919   787446   1225.45   5104333   427.7   100%   60.66   9.604   5.450   -5.474   0.66   2480.9   509.6   2532.7   11.61
189.00   108678   519694   786221   1225.45   5104333   427.7   100%   60.69   9.603   5.462   -5.466   0.66   2486.3   504.1   2536.9   11.46
190.00   108678   518468   784995   1225.45   5104333   427.7   100%   60.73   9.601   5.475   -5.459   0.66   2491.8   498.6   2541.2   11.32
191.00   108678   517243   783770   1225.45   5104333   427.7   100%   60.76   9.600   5.487   -5.451   0.66   2497.3   493.2   2545.5   11.17
192.00   108678   516017   782544   1225.45   5104333   427.7   100%   60.79   9.598   5.500   -5.444   0.66   2502.8   487.7   2549.9   11.03
192.30   108678   515650   782177   1225.45   5104333   427.7   100%   60.80   9.598   5.504   -5.441   0.67   2504.4   486.1   2551.2   10.98


Pasting spreadsheets to forums is not good.  Anyway this in A11 and the velocities are space fixed.
For reference:
161.63 OESC S1
162.3 Stage separation.
164 ESC S2.
You can see the acceleration drops from 2.76 m/s at OESC to Earth's gravity.  That continues for ~ 2 sec. Then S2 ESC.  In this scenario the total velocity meets/exceeds first stage total velocity in ~3.5 sec. 161.63-165.  Note most of the acceleration is going into horizontal velocity also Earth's gravity effect is diminishing.  I realize that the trajectories will be different so a one to one comparison is not valid.
Note that the speadsheet averages every one second(or one row) but it is a close approximation.  BobB's sheet and mine are very similar, he helped a lot and I give him credit for sticking with me during the development.
Note also the total velocity never decreases, just slows from the vertical gravity effect, horizontal was only slowed by the atmosphere.

That is  very interesting Mr. Knight. Looks like a lot of work went into that.

IIRC, Saturn V staging involved retro-rockets (eight IIRC, fitted at the base of each of the outer F-1 engines - two each engine). They were fired for about a second, and actually decelerated the first stage during staging. Same again for second stage staging using four solid-fueled retro-rockets located in a conical adapter at the front end of the second stage.  They did this because they were worried about the discarded stages running into the back of the next stages as they slowed down after engine cut-off. Clearly, stages slowing on MECO and SECO was an issue that needed to be dealt with. The Russians dealt with it by using open frameworks to connect stages and hot starting the stages.

The question for me is, does that slowing down impact the dynamics sufficiently to affect payload in any significant way?

You are correct, except for A15, when they reduced the number to four to attempt to save weight as they were carrying the rover for the first time.  S2 more or less destroyed S1 when ESC occurred, because it did not slow as much as the engineers/computers projected.  The four where re-installed in both A16 and A17, IIRC.

[bknight]

Another thought....how does Starship manage ullage prior to vacuum Raptor ignition? Hot staging removes the need for any ullage motors (or firing of the reaction controls for ullage purposes), so their is a weight saving there.

I asked that over at CQ and IIRC the answer was hydraulic/mechanical.

[Zakalwe]

Another thought....how does Starship manage ullage prior to vacuum Raptor ignition? Hot staging removes the need for any ullage motors (or firing of the reaction controls for ullage purposes), so their is a weight saving there.

I asked that over at CQ and IIRC the answer was hydraulic/mechanical.

Interesting! How does that work? On the Falcon 9 there is a "pole" that reaches up into the bell of the 2nd stage motor. I believe that it uses pneumatics to stage, presumably compressed gas to actuate a piston that shoves the stage away from the booster.

[Peter B]

Well, the hot staging seemed to work.

I switched it on barely a minute before launch, and I think it was a pretty impressive sight. A shame that both stages failed, but the launch as a whole worked a lot better than last time.

[bknight]

Well, the hot staging seemed to work.

I switched it on barely a minute before launch, and I think it was a pretty impressive sight. A shame that both stages failed, but the launch as a whole worked a lot better than last time.

For the second stage, but not much for the first stage.  I agree this launch was much better than the first, all the engines ignited and lifted the vehicle off the pad much faster than in April.

[JayUtah]

At this stage of development (pun intended), that was an impressive flight.

[Zakalwe]

A very impressive sight!

The booster flip looked very energetic. I wouldn't be surprised if the motors ingested gas at start-up hence the RUD. Musk reckons that they could be ready of another flight in 3-4 weeks Musk time notwithstanding). I bet that the next one will be another step forward.