Is it possible for two moving objects to be at rest with respect to each other?

Answer 1

Fasten your seatbelt, cause you've asked one of "those" questions. Fortunately we can handle it. Ready? Let's go. There are two kinds of collisions. One is elastic and the other is inelastic. We're going to set aside elastic collisions here because this is Mechanics 101 of Physics course work. With that in mind, let's just look at inelastic collisions. In inelastic collisions, no energy is "lost" in the deformation of the objects involved. Let's keep it really simple and look at those solid steel balls of uniform size suspended on strings. You sometimes see them on an executive desk as a "toy" of sorts. Note that we have the balls suspended in air, the suspension system, and gravity. That's our system, and it is important to have a closed system to do an accurate evaluation. As regards those balls on strings, if you pull one back a bit and release it, it will strike the others. The result will be that one ball will come off the other end obeying the law of conservation of momentum. Repeat the experiment with two balls, and two come off the other end. Not one ball moving "twice as fast" as the two that were swung into the motionless balls, but two balls. Momentum is still conserved. The energy is accounted for. If we reduce the experiment to have a two-ball setup, the swinging ball should come to rest as it bumps into the lone ball. And that lone ball will be accelerating to the velocity the other ball had when it struck. As long as a gentle swing is given, there will be no energy lost in the collision. We're keeping the collisions inelastic. The balls should be "clicking" and alternately accelerating each other until air friction and a bit of loss in the suspension mechanism bleeds off the energy. As regards both balls being at rest after a collision, you will not see that because the energy must go somewhere to be conserved. If you lift both balls the same distance away from each other and release them at the same time, they will come to rest after the collision. This is because the energy of one ball is equal to and has an opposite vector or direction as the other ball's energy. If a differential in the mass of two objects is encountered, both objects will be moving after the collision as the smaller one will "bounce off" the larger one to some extent. But the larger one will move. The larger the difference in mass, the less the larger one will move, just as you might expect. But as long as the objects are free to move, both will move. The energy must be accounted for. It has to "go somewhere" in any action. We cannot get both objects to be at rest after a moving object strikes a stationary one if the objects are free to move. If one object is not free to move, then we have to look at what is holding it in place as part of our system. Let's change our experiment. A block of steel a meter square is sitting on a table and one of our little steel balls is hung on a wire beside it. Swing the ball gently into the block of steel, and the ball bounces off. The steel will not move because friction with the table will not let it. The ball comes away with most of the energy it hit with, and a bit will be "lost" to the cube and table. They don't move in this case but "absorb" the energy that might have caused the cube to move if it could have. Links are provided.

Answer 2

Yes.

Consider both the objects to be moving with the same velocity in the same direction. Then, the velocity of one of the objects with respect to the other is zero and vice versa, i.e., relative velocity is zero.

If the relative velocity is zero,i.e., one object appears to have zero velocity (no velocity) with respect to the other, then they are considered to be at rest with respect to each other.