I'm currently reading Understanding Physics by Asimov and am stuck on an example he gives regarding conservation of momentum.

What I don't understand is, why half the velocity? -2v makes sense to me if the momentum of the small mass "canceled out" an equal momentum of the larger mass, but -1.5v doesn't make sense. I'm obviously missing something, I just don't know what. Thanks!

If the new velocity is -2v, then the new momentum is (4m)(-2v)=-8mv,right? But how could it have more momentum than initially given unless an external force acts?

But if no external force acts, then the momentum must be the same as it was initially given.
Initial momentum is -2mv. So if the new mass is 4m, then the new velocity must be -v/2 so that the new momentum is (4m)(-v/2)=-2mv.

Momentum is a vector so it can be + or - depending on direction.
p=mv m is mass in Kg and v is velocity (vector) in ms^-1 so p is momentum in Kgms^-1.
If no units are given then its just the same except it would be something like u ms^-1.
p = mv = 1*3 = 3
p = mv = 3*-3 = -9 3+-9 = -6
v = -6/4 (add up momentum and masses) = -1.5ms^-1
I think this is correct. The thing to rember is that momentum is always conserved.

The momentum of the small mass does "cancel out" an equal momentum (in the opposite direction, of course) of the larger mass. The total momentum is now -2mv. To find the new speed you must divide by the mass of the entire system, which is now 4m. So: Vf = (-2mv)/4m = -v/2, in other words: half the original speed v.