Why cannot we go in outer space with constant speed? Sorry if this question is very childish for experts here. This was the question I had in school and years later it has resurfaced. I did some google search but getting the same equations of gravitation. 
So I have learned that to escape the gravitational field of Earth and go in outer space the rocket has to be accelerated greater than the gravitational acceleration of Earth i.e. 9.8 m/s2. 
I am not able to understand as much as I think if I just send a rocket in an upward direction and keep it propelling with a constant velocity even as low as 1 m/s will it not escape gravity and go in outer space. Why does the acceleration constraint there?
 A: You CAN go at a constant speed. 
In fact, if you apply a force upwards that would make a rocket accelerate at $9.8 m/s^2$ in empty space, when not under the influence of gravity, it WOULD go upwards at a constant speed! 
The point is that since the earth's gravity pulls back on objects with that acceleration of $9.8 m/s^2$ (close to the earth's surface at least) you must give that acceleration upwards in order to counteract the earth's gravitational pull and move upwards at a constant velocity, and must give more than that acceleration to actually accelerate a rocket upwards.
A: I think you are getting confused about the concepts of velocity, acceleration, and force.  If you can somehow maintain a 1 m/s velocity component away from the surface of the earth you can get to "outer space." (There are circular components of velocity that 
real spaceships have to allow for, too. Just trying to address the radial, or to-and-from motion.)
The problem is that there is a gravitational force, $\vec{F_g}$, always pulling your ship back toward earth. To initially get to 1 m/s you must provide a thrust force in the direction opposite $\vec{F_g}$ and greater than it so that the net force is away from earth. Once you get to 1 m/s (or any other speed you want to consider), you still must provide an upward force or gravity will pull the ship back unless your speed is what is called escape velocity. exactly how to calculate that is beyond the scope of your question, but for earth starting near the surface it's approximately 11 km/s.  If you turn your engines off before reaching that speed, you'll not escape earth, but you might orbit.
If you keep providing thrust and maintain an outward velocity of any value, you can escape.  The problem with maintaining thrust is carrying enough fuel to do that. Fuel has mass, and you have to burn fuel to carry fuel with you for burning later. Eventually the fuel is gone and the thrust is gone.
