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Here's how I understand it: A horse is hitched to a cart. The cart will move if the horse is able to move, and for the horse to move, the force that it exerts on the cart should be greater than the force the cart exerts on it.

But the way I look at it, it is not possible for the horse to pull the cart with a greater force than the cart pulls it in the other direction. The horse gets all his force from the ground, right? It pushes the ground backward and the ground pushes it forward. Now, here's where I think you'll find the mistake that I'm making, but I can't convince myself that it is a mistake (I'm not being willful!); for the force with which the horse is being pushed forward should also be the force with which it pushes the cart forward and consequently, the force with which the cart pulls it backward. I know that this is not what Newton's third law says and this is not the correct action/reaction pair, but I can't think of it any other way. To me, the horse's role is merely to transmit the force it is being pushed forward with by the ground over to the cart. So, let's say, the ground pushes horse with 20 N, the horse pulls cart with 20 N, the cart pulls horse with 20 N. If the force that the ground pushes the horse with is doubled, the force on the horse by the cart and on the cart by the horse should also be doubled and we are still left with 0 net force. I understand that this is not a requirement of Newton's Third law, but in this particular case, I can't make myself believe that it is not so.

I hope you understand what I'm having trouble understanding, and once again, I'm sincerely very sorry.