What magnitude of force does a ball of mass 0.75 kilograms need to be hit so that it accelerates at the rate of 25 meters/ second^2

Force = (mass) x (acceleration)

           = (0.75 kg) x (25 m/s²)  =  18.75 newtons

                                           (about 3.9 pounds)

You don't "hit" with force.  You apply or exert force. 

As long as 18.75 newtons (3.9 pounds) of net force continues to act
on the ball, the ball accelerates at 25 m/s².  The instant the force stops,
the acceleration stops.

The reason we think of hitting as such a violent, instantaneous event
is that the force on the ball lasts for such a short time.  In that time,
a large amount of momentum needs to be transferred to the ball,
so it takes a large impulse (force x time). 

From another angle:  In that short time, we want the ball to acquire a
large speed, so the acceleration has to be huge, which requires a huge
force during that short contact time.

Example: 
If the pitcher pitched a fast-ball to you at 100 mph, and you want it to leave
your bat at 100 mph in the opposite direction, then you need to change its
velocity by 200 mph (89.4 m/s).  But your bat is only in contact with the ball
for, say, 1/10 of a second, so that's all the time you have to change the
ball's velocity.

 Velocity = (acceleration) x (time)
     
 Acceleration = (velocity) / (time)  =  (89.4 m/s) / (0.1 sec) = 894 m/s²

 Force = (mass) x (acceleration)

           0.75 x 894  =  670 newtons

                              =  151 pounds, for 1/10 of a second