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If one equipped the SRB like the ones used with the Shuttles with minimal attitude control systems and launched it as a standalone rocket with no payload - what speed would it reach? Would it be capable of entering the Low Earth Orbit?
More generally, what is the viability of (staged or not) SRB-only launch vehicles?
No, a quick calculation yields a $\Delta v$ of about 4.6 km/s and you need about 9 km/s to get to low-Earth orbit. You'll lose a lot of that velocity to aerodynamic drag as well as the vertical portion of the flight, so a rough estimate of the final speed at burnout would be somewhere around 3 km/s -- or about 10,000 km/h.
Using the specifications from the SRB Wikipedia article (which admittedly have some conflicts):
$m_{full} = 590000 kg$
$m_{empty} = 91000 kg$
$I_{SP} = 250 s$ (242 at sea-level, 268 in a vacuum, most of the flight is in the atmosphere so we'll round up to 250)
$\Delta v = g_0 I_{SP}\ln \frac{m_{full}}{m_{empty}}$
Here is a simulation of a launch composed of vertical flight until reaching 100 m altitude, then pitching over to a constant pitch of 85 deg until reaching 1000 m altitude, then reverting to a gravity turn at 0 deg angle-of-attack. This sends the SRB to an altitude over 500 km, with a peak speed around 3.7 km/s at burnout.
Here is a look at the pitch, flight-path angle (direction of motion), and angle-of-attack (difference between pitch and flight-path angle) as a function of altitude. Although it might seem strange that the rocket starts pitching over after only 100 m of flight, this is necessary to start gaining enough velocity to reach orbit -- the rocket is still climbing throughout most of the flight.
Note that I'm using a simple atmospheric model for the air density:
$\rho = 1.225 e ^\frac{h}{8000}$
And a constant drag coefficient of $C_D = 0.8$:
$D = \frac{1}{2} \rho v^2 C_D A$
Mass flow rate is assumed to be constant, computed from the stated sea-level thrust (12 MN) and specific impulse (242 s), whereas specific impulse is computed based on a simple mixture rule:
$I_{SP} = I_{SP,sea} \frac{\rho}{\rho_{sea}} + I_{SP,vac}\left( 1 - \frac{\rho}{\rho_{sea}}\right)$
Then thrust is computed from the specific impulse and mass-flow:
$T = \dot{m} g_0 I_{SP}$
Note that the equations of motion in 2D are:
$\ddot{r} = -\frac{\mu}{r^2} + r \dot{\theta}^2 + \frac{T}{m} \sin \phi - \frac{D}{m} \sin \gamma$
$\ddot{\theta} = -\frac{2 \dot{r} \dot{\theta}}{r} + \frac{T}{m r} \cos \phi - \frac{D}{m r} \cos \gamma$
Where $r$ and $\theta$ are the polar coordinates about the Earth's centre, $m$ is the vehicle mass, $\phi$ is the vehicle pitch and $\gamma$ is the flight-path angle (velocity direction) -- both of those angles are with respect to the local horizon.
As for all-solid fuel launch vehicles, there are a number of them that have been and are still in use (such as the Minotaur family). But, as LocalFluff pointed out, they are usually only sensible for smaller payloads. Lunar Prospector was launched to the moon on an all-solid Athena II although it looks like the upper stage used hydrazine (not solid, but not a complex liquid motor either).
The Ares I-X was a modified STS SRB with with an upper-stage simulator. So far as I know, this was the only standalone launch of SRB hardware. Even the original SRBs were not flight tested before manned launch.
The Ares I-X flight had an apogee of 46 kilometers, less than half the 100 kilometer altitude of the Karman line, which is considered the edge of space and below which orbital trajectories are impossible due to air resistance. Due to the rocket equation we can clearly see that no amount of stacked SRB components would so much as double the altitude, therefore we can conclude that a single SRB could not fly alone to orbit.
User SF. calculates that a staged SRB may be able to launch considerable payload to orbit:
Alone - okay. Staged - quite likely doable. From peak speed of 1.7km/s, and mass of the 3 stages vs 2 stages (from here ) I can find the ISp. Replacing second stage with another such thruster, I'm getting 1.8km/s from stage 1, another 4.9 from stage 2, for a total of 6.7km/s for 172 tons of stage 3. Now if we leave 50 ton of dry mass, and sacrifice 122 tons to SRB fuel (same ISp still!) we're getting another 3.9km/s for a total of 10.6km/s. There, 50 tons to orbit on 3 SRBs.
Here is a video of the launch, complete with on-board camera views:
