Object B is at rest when object A collides with it. The collision is one-dimensional and elastic. After the collision object B has half the velocity that object A had before the collision.

Which object has the greater mass?

There are two types of collision when it comes to the over all conservation of energy comes to play, while the energy of the number of system that is in contact either increases, decreases, or remains intact as per the situation.If the collision is elastic by means that the total energy of the system does not remain the same that it was before but there are some changes which happens inside the total energy that each system posses. And as the whole conservation is all based upon the measurement of mass, m and velocity, v which acts as the product for the momentum,p. And there for we calculate the over all values of the momentum before and after the collision.So, the effect of collision does have the mass,m  of the different bodies to effect on the collision of the two system. The total mass,m of the body does have a greater part in obtaining greater or less velocity for the respective body or any object. Let's say that the mass, m of the of the body A is not in motion and that the body B struck it with a given velocity that it had acquired during its motion then we can say that, the body "B" will have the half as velocity as compared to that of body "A".