What will happen if a force applied to a body moving with a constant speed along a straight line?

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I suspect the confusion arises from choosing reference frames. Let's try different views on the same scenario.

(1)Let the system consist of a block 1 that is sliding on frictionless surface with no external forces acting on it before the collision. (Here I assume that normal force and weight do cancel out, and thus are not included in 'external forces' part.)

When it collides with standing block 2, the forces act on a system. Thus forces are no longer zero and momentum changes.

Very similar scenario happens when we let block 1 stand and block 2 collide:

(2)Let the system consist of a block 1 that is stationary on frictionless surface. Since it is not accelerating, sum of forces acting on it are zero. When block 2 collides with it, forces act on block 1 and momentum is no longer conserved.

(3)Finally consider system of block 1 that is moving on frictionless surface and block 2 that is standing in line of motion of block 1. Assume no external forces act on a system. Therefore: $$\vec F^{ext}=0\implies \vec v=const.\qquad\text{conservation of momentum applies, let mass of block i = }M_i$$ $$M_1\vec v_o=M_1\vec v_1 + M_2\vec v_2$$

Therefore in order to find out how the system behaves, first and foremost we need to define that system.

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