What is the fastest speed an object can acquire when falling down to Earth from outer space?
The answer to your question is called "Terminal Velocity"
This is when the accelerating force and the opposing force cancel each other. In this case it would be gravitational force vs air resistance.
The time at which this will happen depends on the shape of the falling object.
Hence, to find out the terminal velocity the important variable missing is the shape of the object itself.
Ex: a feather reaches terminal velocity as soon as you drop it, a sky diver takes around 20 to 30 sec, a spherical dense object will take much longer.
In this case let's take Newtonian friction factors and assume the object had v0 =0 m/s, at t= 0 and the only acceleration is due to Earth's gravity. Then the terminal velocity can be found using eq. Mg = cv2 where v is terminal velocity and c is coeff of drag. C depends on the shape and the projection of the object.
So, if you know the shape then you can find the C (friction Factor) and using above eq. Find the terminal velocity v.
The fastest speed an object can hit the surface of another object only using gravity is equal to the minimum speed required for an object to escape another one's gravity well, which is called the escape velocity, equal to about 11.186 kilometers per second, the actual maximum speed will be slightly less due to atmospheric friction when it starts to touch the thermosphere, it would likely be burnt into a crisp from hyper sonic heating and extreme deceleration, and will disintegrate if it haven't been.
Objects in space may have just about any speed relative to earth. Some cosmic rays, and other particles, travel near the speed of light.
Many meteors seem to travel at tens of thousands of miles per hour.
Spacecraft orbiting the earth travel at thousands to maybe a couple of tens of thousands of miles per hour.
One way to interpret this question is as follows:
An object is in a geosynchronous orbit at about 23,000 miles (37,000 km) above the surface of the earth.
It is nudged toward the Earth so that it is traveling at 1 mph (1.6 kph).
The object is further accelerated by the Earth's gravity, gaining speed accordingly.
When the object enters the atmosphere, atmospheric drag reduces the speed it would have otherwise acquired to the point that atmospheric drag equals gravitational attraction and the object falls at its terminal speed.
Eventually, it hits the surface of the Earth.
How fast will it be traveling when it hits the Earth?
This could be calculated, except that one thing is missing: aerodynamic drag.
If the object is highly streamlined, like a rocket, it will be falling fast, thousands of miles per hour. If it is not very streamlined, like a leaf, it will be falling slowly, less than one mile per hour.
A comet in a hyperbolic orbit around the sun could plow into the earth at speeds in excess of 70,000 MPH.
The asteroid that struck the earth 65 million years ago was probably moving around 40,000 MPH.
The Stardust return capsule was the fastest man-made object ever to reenter Earth's atmosphere at 28,000 MPH.
Reentry speeds for the Apollo spacecraft were about 25,000 MPH.
The Space Shuttle typically returned to earth at 17,000 MPH.
In 2012, Felix Baumgartner jumped from a balloon at 120,000 feet and during freefall he exceeded the speed of sound (700 MPH).