Question

“A horse pulls on a cart. By Newton’s third law of motion, the cart pulls back on

Answer 1

In order to see what's going on, let's put them in empty space to get rid of any other influences, and let's also make it a push instead of a pull. / / / The horse pushes on the cart, so it begins accelerating away from him. At the same time, because of the equal opposite reaction thing, the cart pushes back on the horse, so the horse starts accelerating backwards, away from the cart. They both accelerate in opposite directions from where they started. BUT . . . their common center of mass doesn't move, and the sum of their momentums (which are in opposite directions) remains zero.

Answer 2

If they are both in space and not on ground, then the above way of thinking is correct. Horse will not be able to accelerate forward.

But here, there are some external forces acting on the horse other than the pull of the cart. That makes the horse accelerate. Cart also gets accelerated.

Horse kicks and pushes the road or ground with his feet backwards. So the ground pushes with an equal force in the forward direction. On the horse, there is a force by the cart, pulling it backwards and a force of friction from the ground and a push forward given by the ground. Pushing forward force by the ground is higher than the sum of others. So horse accelerates.