Step one of any physics problem is to draw a crappy picture in MS paint diagramming the forces acting on each object. Thus,
Now, we know that $\Sigma F = Ma$. Object 1, which is at rest, has an acceleration of 0. Since 0500=0, it has no net force acting on it. That is to say, every force acting on object 1 has a counterforce acting on it in the opposite direction.
Now, let's analyze object 2. It's moving at a velocity of $V_0$ towards object 1. Since it isn't accelerating, there's no net force on the object. This basically sums up the free body diagram.
The center of the diagram shows the collision between the two objects. Collisions are defined by two pretty simple rules:
$$\Sigma p_i=\Sigma p_f$$ $$\Sigma E_i = \Sigma e_f$$
These rules are known as conservation of momentum and energy, respectively. However, they're not very useful in their current forms. We should expand them to fit our needs. 512
I'll assume a perfectly elastic collision for this one. Basically, momentum is solved as $p=mv$, and kinetic energy is solved as $KE=\frac{1}{2}mv^2.
Now, a few things to note about your question:
1. Since you're operating on a frictionless surface in one dimension, any force exerted on object 1 will move it.
2. Force isn't actually related to speed. It's related to change in speed. You're actually moving at a few hundred thousand meters per second right now, believe it or not. And yet you're not feeling any force being applied on you.1
TL;DR
Simply put:
1. Minimum force is anything above 0.
2. Speed necessary is anything above 0.
+ Force is defined as $m*a$ and is not related to speed
+ Force can't be applied instantly, as acceleration is $\Delta v/\Delta t$ and setting $\Delta t$ to zero makes things break.
3. Speed after impact is not the same.
+ It can be used to determine the pre- and post-impact speeds of both objects.
Footnotes
1: er... you don't feel it, but there's some angular forces at work keeping the Earth around the sun and keeping you on the Earth. 343