Can the linear momentum of a system be constant, even though its centre accelerates? My instructor says if the velocity of center or mass is constant, it means that the linear momentum of a body is conserved. So if no external force acts on a body, there is no change in the linear momentum of the system. 


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*Now, if I know the linear momentum of a system to be a constant, does it not imply that the center of mass of the system doesn't accelerate? 

*My instructor thinks otherwise. Where am I going wrong?
 A: It's not difficult to show that the total linear momentum of a system can be found by multiplying the velocity of the center of mass by the total mass. If the momentum is constant (in the absence of a net external force),  then the velocity of the center of mass is constant. But be careful with your statements.  The center of mass of an object (or a system) may not be at its “center”.  If you throw a ball which has a non-uniform mass distribution, the center of mass will follow a smooth curve, but the center of the ball may revolve around that.
A: Consider the center of mass of a body and then any point P on the body. Then in your frame of reference, you saw the C.M accelerating. Obviously, point P can't remain in the same position of space,else OP distance will change. Thus you must see it accelerate with the same linear acceleration as that of the center of mass. But, at the same time it can also follow a rotational motion, such that OP remains same as the before. However,the particle must accelerate. Same will apply for all particles of the body.
The thing is that individual particle's acceleration might not be equal to that of the center of mass. For eg in rolling without slipping, the CM has constant linear acceleration but each particle has different acceleration at different instants, both in magnitude and direction.
A: The above statement on conservation of linear momentum refers to the law of inertia in a specific frame of reference.
Here is the explanation of the scenario where a body can have some acceleration, but still have constant linear momentum:
Imagine a person riding a roller-coaster in an amusement park.
When the roller-coaster is considered as the frame of reference, the rider sitting in a seat is stationary, and hence will have a constant linear momentum, in reference to the roller-coaster, or the fellow riders.
However, when the roller-coaster is moving, the rider will also have the same acceleration as that of the frame of reference (roller-coaster).
For an observer on the ground, the person riding on the roller-coaster has an acceleration.
