QUESTION: 
How fast can a rocket travel in 1 minute?

ANSWER from Mike Wilhoit on October 3, 1997:
Well, it depends on what kind of rocket you are talking about.  But for
example the Shuttle, at main engine cutoff, is traveling at about 26,000
feet per second.  In one minute at that speed, it will have flown almost
300 miles, or about 475 kilometers!

ANSWER from Jim Draus on October 3, 1997:
The speed of a rocket is directly affected by 2 factors.  The first is
mass ratio, which is a ratio the weight of the rocket at lift-off
compared to the weight of the rocket at engine shutdown. The second
factor is specific impulse, which is the amount of thrust produced from
each pound of propellant per second.  The higher the mass ratio and the
higher the specific impulse, the faster the rocket can go.

In reference to the Space Shuttle, and in answer to your question, based
on the Orbiters mass ratio and its propulsion systems' specific impulse,
the Orbiter reaches an on orbit velocity of 17,322 mph which is 289
miles per minute.

ANSWER from Jim Smith:
During the first 90 seconds of flight the flight control systems provide
load relief by making adjustments to reduce vehicle loads at the expense
of maintaining a precise trajectory.

To keep dynamic pressure on the vehicle below a specified level, on the
order of 580 pounds per square foot (Max-q), the main engines are
throttled down at approximately 26 sec. (to around 60%) and throttled back
up at approximately 60 sec.(term go for throttle up 104%).  This also
reduces heating on the vehicle.  Because of the throttling at this time
the term "thrust bucket" evolved.  Your question "How fast can a rocket
travel in 1 minute?" can be answered two ways.

Remembering what was read from the above paragraphs.   The Shuttle is at
the end of the thrust bucket.  I contacted the flight director's office
and they quoted me this response.  The Shuttle is approximently going Mach
2 or 1480 mph @ the 60 second mark.  This varies with weight and
trajectory. Second To answer this, I would use the simple rocket equation:
Obtained from a fellow Shuttle on Line participant William R. Britz Rndz
Flight Dynamics Officer.

Wp = Wi * (1 - e**(-dV/g*ISP))

Where, 

Wp = weight of propellant expended during thrust arc
Wi = initial weight of the vehicle
dV = delta velocity change
g = gravitational acceleration (32.174 ft/sec^2)
ISP = specific impulse of the engine in use.

Solving for dV, the equation becomes:

dV = -g*ISP*ln(1- Wp/Wi)

In order to use this equation, you must know some details about the 
vehicle and its rocket engine.  For example, one of the Shuttle's OMS 
engines has the following properties: 

ISP = 315.0 secs
Weight loss rate = 19.24 lbs/sec
(Thrust = 6060.0 lbs)

The shuttle itself weighs about 220,000 lbs.  Assume 220,000 is the 
starting weight (Wi), then in 60 seconds, the weight of propellant 
expended is 60*19.24 = 1154.4 lbs. Now plug this into the rocket equation
above to find the change in  velocity due to a 60 second single OMS burn
firing:

dV = -32.174 * 315 * ln(1 - 1154.4/220000)
dV = 53.32 feet/sec

This means that a 60 second OMS burn will CHANGE the shuttle's current 
velocity by 53.32 feet/sec.  On orbit, the Shuttle "coasts" at about 
25,000 feet/sec.  A 53.32 foot/sec delta-v change for the shuttle would 
be large, and could change one side of its orbit by ~30 nautical miles!

Hope this is helpful.