Suppose you have a 136-kg wooden crate resting on a wood floor. what maximum force (in n) can you exert horizontally on the crate without moving it? if you continue to exert this force (in m/s2) once the crate starts to slip, what will the magnitude of its acceleration then be?

-- The maximum horizontal force without moving the crate
depends on the friction along the surface where the crate
meets the floor. 

That force will be greater if the crate and the floor are both lined
with sandpaper, and it will be less if there is a layer of oil between
them.  We don't know anything about those surfaces.

If ' μ ' is the coefficient of static friction between the crate and the floor,
then the force of friction acting opposite to a horizontal push is

( μ )·( mass )·( gravity )

= ( μ )·( 136 kg )·( 9.8 m/s² )

=  1,332.8μ Newtons.

A push less than this will not move the crate.


-- " m/s² " is a unit of acceleration, not a unit of force.

Once the crate starts to slip, its behavior depends on the coefficient
of KINETIC friction between it and the floor, which we also don't know,
but this number is usually smaller than the static coefficient.  So the
force of friction will decrease once the crate starts to slip, and you're
correct in expecting that it'll accelerate in the direction of the push,
even though the strength of the push doesn't change.  Again, we
can't estimate the magnitude of the acceleration without knowing the
nature of the friction where the crate meets the floor.