(Or, if you want to treat them as a composite object, keep in mind that it's not a rigid body.) So it's not that they cannot accelerate - these two forces don't cancel out, as they aren't acting on the same object. These two forces act on separate objects, and are directed at one another. When the astronaut pulls on the toolbox (via the tether), the toolbox will also pull on the astronaut. The astronaut is freefloating outside of the International Space Station, doing some maintenance work. Replace the horse with an astronaut, and replace the cart with something heavy - maybe a toolbox of some sort. Let's change the situation a bit, and change the setting to - space. Since the two objects are attached together, they are technically the same object, and they cannot accelerate According to Newton's third law, when the horse pulls on the cart, the cart will also pull backwards on the horse. This process repeats over and over again and the (horse + cart) system is moving relative to the earth. First the horse does a momentum exchange with the Earth, then it does a momentum exchange with the cart. The reason why your original point doesn't work is because in some cases the system consists only of the horse and the cart, and sometimes it consists of the horse and the Earth. Now bring the rope's length down to essentially zero and consider this chain of events being repeated rapidly. The horse will appear to be moving relative to the Earth, but keep in mind that the Earth has been pushed in the opposite direction. Since momentum has to be preserved, however much momentum the horse imparts to the Earth, the Earth will push back. Now if the horse burns some of the grass it ate and uses it to generate some energy which it then transferred to the earth using its legs, we would need to include the Earth in the momentum system. So if the cart has 2x the mass of the horse, it will be moving at 45% of the horse's speed and in the same direction because the horse pulled on the rope, the rope pulled on the cart, accelerating it, the cart pulled back on the rope, and the rope pulled back on the horse, stopping it. If we can assume that the horse and the cart represent a closed system, then if the horse comes down to 10% of its original speed, the velocity of the cart will be the horse's decrease in velocity multiplied by the ratio of the horse's mass to the cart's mass. Newton's Laws basically evaluate to the following statement: Momentum is preserved in any system that receives a zero outside force, and the change in momentum of any system is the change in the product of the mass and (directional) velocity of the system. So the cart does indeed pull on the horse, but in doing so receives some (probably most) of the horse's momentum. However much momentum the horse had when it hits the end of the rope will be transferred to the cart. When the rope runs out, the horse will most likely get pulled almost to a stop in what will be a very dramatic surprise for the horse! But the cart will start moving. The horse is allowed to run away as fast as it can. The best way to picture this is to consider a horse attached to a cart by a 100m rope coiled up. You are correct in suggesting that the cart pulls back on the horse.
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