Math teacher in over his head

Main Question or Discussion Point

Math teacher in over his head :)

Not so much "over my head" as I don't have as much time as I'd like to research the answers. My 7th grade honors class is doing a project on "How fast is the Space Shuttle in terms of Mach?" (My title, not theirs :) ).

They didn't have too much trouble finding the speed of sound at various altitudes, but the math to find the shuttle velocity at those altitudes (and how fast it got there) is WAY beyond them and I've got too much on my plate to review my college physics :( So, will some enterprising soul help me fill in the table below...?

Altitude / SST velocity (any reasonable units) / t+
5,000
10,000
15,000
20,000
25,000
30,000
35,000

TIA :)

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Q_Goest
Homework Helper
Gold Member
There's some information on the NASA web site, but not enough to fill out your chart. Here's what they have:

In the transition phase, the angle of attack continues to ramp down, reaching the approximately 14-degree angle of attack at the entry Terminal Area Energy Management (TAEM) interface, at approximately 83,000 feet altitude, 2,500 feet per second, Mach 2.5 and 52 nautical miles (59 statute miles) from the landing runway. Control is then transferred to TAEM guidance.

During the entry phases described, the orbiter's roll commands keep the orbiter on the drag profile and control cross range.

TAEM guidance steers the orbiter to the nearest of two heading alignment cylinders, whose radii are approximately 18,000 feet and which are located tangent to and on either side of the runway centerline on the approach end. In TAEM guidance, excess energy is dissipated with an S-turn; and the speed brake can be used to modify drag, lift-to-drag ratio and flight path angle in high-energy conditions. This increases the ground track range as the orbiter turns away from the nearest Heading Alignment Circle (HAC) until sufficient energy is dissipated to allow a normal approach and landing guidance phase capture, which begins at 10,000 feet altitude. The orbiter also can be flown near the velocity for maximum lift over drag or wings level for the range stretch case. The spacecraft slows to subsonic velocity at approximately 49,000 feet altitude, about 22 nautical miles (25.3 statute miles) from the landing site.

At TAEM acquisition, the orbiter is turned until it is aimed at a point tangent to the nearest HAC and continues until it reaches way point 1. At WP-1, the TAEM heading alignment phase begins. The HAC is followed until landing runway alignment, plus or minus 20 degrees, has been achieved. In the TAEM pre-final phase, the orbiter leaves the HAC; pitches down to acquire the steep glide slope, increases airspeed; banks to acquire the runway centerline and continues until on the runway centerline, on the outer glide slope and on airspeed. The approach and landing guidance phase begins with the completion of the TAEM pre-final phase and ends when the spacecraft comes to a complete stop on the runway.

The approach and landing trajectory capture phase begins at the TAEM interface and continues to guidance lock-on to the steep outer glide slope. The approach and landing phase begins at about 10,000 feet altitude at an equivalent airspeed of 290, plus or minus 12, knots 6.9 nautical miles (7.9 statute miles) from touchdown. Autoland guidance is initiated at this point to guide the orbiter to the minus 19- to 17-degree glide slope (which is over seven times that of a commercial airliner's approach) aimed at a target 0.86 nautical mile (1 statute mile) in front of the runway. The spacecraft's speed brake is positioned to hold the proper velocity. The descent rate in the later portion of TAEM and approach and landing is greater than 10,000 feet per minute (a rate of descent approximately 20 times higher than a commercial airliner's standard 3-degree instrument approach angle).

At 1,750 feet above ground level, a pre-flare maneuver is started to position the spacecraft for a 1.5-degree glide slope in preparation for landing with the speed brake positioned as required. The flight crew deploys the landing gear at this point.

The final phase reduces the sink rate of the spacecraft to less than 9 feet per second. Touchdown occurs approximately 2,500 feet past the runway threshold at a speed of 184 to 196 knots (213 to 226 mph).
Ref: NASA

You might consider asking the experts. Bet the kids would enjoy emailing astronauts.

Gokul43201
Staff Emeritus
Gold Member
This is purely theoretical but for a pretty good approximation, you could use v^2 = GM/(R+h)

v = orbital velocity,
G = univ. grav. const.
M = mass of earth
R = mean radius of earth
h = orbital height

Q_Goest
Homework Helper
Gold Member
Ion: I'm assuming the altitudes you provided above are in feet or perhaps meters. Perhaps you could clarify for everyones sake.

pervect
Staff Emeritus
Is the problem estimating the shuttle's speed on it's upward trip, it's downward trip, or both? Most of the solutions presented seem to be oriented towards figuring out it's speed on the downward trip, though the web page that talks about the glide approach also talks a bit about the ascent phase, too.

Q_Goest said:
Ion: I'm assuming the altitudes you provided above are in feet or perhaps meters. Perhaps you could clarify for everyones sake.
Sorry. Altitude is in feet. As much as I'd like to do metric, for 7th graders, it seems easiest to use feet (they can grasp that concept better).