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According to Newton's Second Law, consider the case of wanting to accelerate the mass of 3 kg by 3 m/s^2 on a frictionless surface, I need to exert 9 N of force. My question is that

  1. To let the mass accelerate forever, do I have to exert the force of 9 N forever or just for a short amount of time? What's the difference?
  2. If I push the same object quickly on the surfacet that has a kinetic friction of 1 N and the object starts accelerating with 3 m/s^2, how will the friction cause the acceleration to change over time?

These kind of things really confuse me. What my teacher teaches is only what will happen immediately after exerting the force but not what will happen after that.

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Perhaps you are confusing velocity and acceleration. A resultant force applied for some time causes an increase in velocity. Acceleration measures how fast the velocity is increasing. If you want velocity to keep increasing (acceleration) you must continue to apply the force. If you stop applying the force the velocity will stop increasing. The acceleration will return to zero but the velocity will remain what it was when the force stopped.

In your 2nd example if you apply the same force of $9N$ on the $3kg$ object while kinetic friction is $1N$ (in the opposite direction) you will not get an acceleration of $3m/s^2$. The acceleration is caused by the resultant force not the applied force. In this case the resultant force is only $8N$ so the acceleration is $\frac83 m/s^2$. The acceleration is different from the case without friction, but as before it does not change as you continue pushing unless the resultant force changes. If you stop pushing the object continues moving so there is still kinetic friction which slows it down until it stops. The resultant force is now $1N$ so the acceleration is $-\frac13 m/s^2$. The minus sign indicates that velocity is decreasing.

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  1. If there's only a single force of 9N on the mass, then you must apply the force forever. The difference is an impulse force creates an impulse acceleration while a constant force creates a constant acceleration.
  2. If the kinetic friction acts over the lifetime of the movement, friction will act in the direction against the movement of the object. Effectively, it seeks to reduce the acceleration of the mass which will slow the object down.
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