I recently took a look at this little data sheet by NASA: Math and Science @ Work; Space Shuttle Ascent

Time Altitude Velocity Acceleration Comment
(s) (m) (m/s) (m/s^2)
20 1244 139 18.62
40 5377 298 16.37
280 105321 2651 13.92
300 107449 2915 14.90
320 108619 3203 15.97
340 108942 3516 17.15 maximum altitude
360 108543 3860 18.62
380 107690 4216 20.29
400 106539 4630 22.34
420 105142 5092 24.89
440 103775 5612 28.03
460 102807 6184 29.01
480 102552 6760 29.30 maximum acceleration
500 103297 7327 29.01
520 105069 7581 0.10

original screenshot

Note: Notice from the table that the altitude was negative at liftoff. Zero altitude can be described as a specific distance from the center of the Earth. Since the Earth is not perfectly spherical the location of the launch just happens to be below this specified point. Also, because this is a calculated number, some degree of error may be present.

As you can see, as time passes, the velocity and altitude of the rocket went up exponentially. [CORRECTION: I was mistaking this data for something else. The vel/alt does NOT go up exponentially, sorry about that!] However, why did the altitude stop increasing after reaching 108,000m, and then decrease after that?

Also, why was the acceleration rate inconsistent throughout the takeoff? Why was it the highest when the altitude was decreasing?

Thank you, I'm working on a project related to the Rocket Equation and am wondering if this is worth mentioning in said project. :)

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    $\begingroup$ "why does the velocity basically "stop" after reaching 108,000m". Only the vertical component of the velocity. The total velocity shown in the third column is still increasing. The target altitude seems to be 105km. But, the first time the vehicle hits that target, the total velocity is only 2600 m/s. It is far far below the required velocity of 7200 m/s required to stay in orbit. $\endgroup$ – AJN Jun 16 at 10:07
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    $\begingroup$ "go up exponentially" is wildly misleading and inaccurate. The curve is, indeed, exactly unlike "exponentially". $\endgroup$ – PcMan Jun 16 at 13:42
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    $\begingroup$ Partial answer: to "why is the acceleration rate inconsistent throughout the takeoff? Why is it the highest when the altitude is decreasing?" the acceleration increases as thrust remains constant while vehicle mass is decreasing due to used fuel. (thrust actually increases a little bit as atmosphere thins, this would not be obvious in your dataset). The abrupt drop in acceleration at 2 minutes is when the thrust drops enormously, due to the Solid Motors completing their burn. $\endgroup$ – PcMan Jun 16 at 13:45
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    $\begingroup$ what-if.xkcd.com/58 "getting to space is easy. The problem is staying there.... To avoid falling back into the atmosphere, you have to go sideways really, really fast." $\endgroup$ – Mooing Duck Jun 17 at 16:14
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    $\begingroup$ The high-altitude altitude drop-off was informally referred to as "droop," IIRC. The trajectory was shaped this way for performance optimization reasons (reflecting tons of analysis and "big physics"). An interesting independent question would inquire about the specific reasons behind said "droop." $\endgroup$ – Digger Jun 17 at 17:19

The drop in acceleration around 40s into the flight is the shuttle throttling down to reduce the aerodynamic load on the vehicle. It then accelerates when past this point.

The drop in acceleration at 2 mins into the flight is due to the solid rocket boosters running out and being discarded.

Acceleration then continues to build, as the thrust from the engines is constant, but the vehicle mass gets less and less as the fuel is consumed. The peak acceleration is due to the low mass when the tank is almost empty, and not related to the altitude.

In fact, towards the end of the ascent the shuttle is throttling down as the mass goes down, to keep the acceleration below 30m/s/s, for structural reasons.

The change in altitude is due to the shuttle overshooting its initial altitude, so it can put all its thrust into horizontal velocity after that. So it actually falls a little as it continues to burn and accelerate horizontally, but eventually this horizontal velocity results in it not falling anymore (as the earth falls away below it at the same rate).

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    $\begingroup$ OP is asking two separate questions, and although I think the acceleration aspect is covered here, the why of the altitude loss is poorly explained. It's fully possible to fly an ascent profile which never has any downward velocity. I suspect there is a complicated reason lurking behind the choice of profile, and would welcome further info. $\endgroup$ – Innovine Jun 17 at 15:35
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    $\begingroup$ It has to do with PEG guidance achieving the velocity target; it's explained on pages 4-5 through 4-12 of the Ascent Guidance Workbook, but I'd be lying if I said I understood it. gandalfddi.z19.web.core.windows.net/Shuttle/… $\endgroup$ – Organic Marble Jun 17 at 21:45

For what it's worth, the Saturn V did the same thing getting to parking orbit. Different vehicles with different propulsion and staging, but the outcome is essentially the same:

Saturn V altitude

Source: Saturn V Flight Manual. Note that the graph labels for the second (S-II) and third (S-IVB) stage cutoffs are incorrectly swapped. The source explains the third stage's burn as an insertion into the Earth parking orbit:

The S-IVB first burn inserts the vehicle into a 100 nautical mile (NMI) altitude, nearly circular, EPO.

Innovine's answer provides one explanation: the vehicle has surpassed the desired altitude, but needs to gain speed to stay in orbit. Another way to look at it is in terms of apogee and perigee. The vehicle has reached maximum altitude (apogee), but the perigee intersects the ground. Another burn is necessary to raise the perigee to orbital altitude.

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    $\begingroup$ PEG guidance again. $\endgroup$ – Organic Marble Jun 18 at 12:08

Let's look at the numbers. The acceleration is high at the start (full force!). It drops a bit at 40 seconds but then increases steadily due to mass loss. Between 100 and 120 seconds, there is a sudden drop in the acceleration. The two main thrusters are thrown off. From the lower value the acceleration increase continuously. In a somewhat linear fashion. When the highest altitude is reached the shuttle moves horizontally after which it moves toward a lower orbit. The acceleration is increasing towards a max around 480 seconds. Around 500 seconds the force is turned off and the shuttle has reached a height of about 105 kilometers. This height will increase until a stable orbit is reached.

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    $\begingroup$ This answer does not add anything new beyond the other answer. Once the main engines are shut down the orbit does not change unless another propulsive maneuver is performed (or hit the atmosphere). $\endgroup$ – BrendanLuke15 Jun 17 at 11:26

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