The horse and the cart

Problem

A horse refuses to pull the cart, saying: “By the third law, whatever force I apply to the cart, the cart applies an equal and opposite one on me. The resultant is zero, so the cart will never move!”. Take apart the horse’s reasoning by identifying which forces act on which body.

Solution

The horse makes the classic mistake in using the third law: adding together two forces that act on different bodies. Action and reaction do indeed have equal magnitude and opposite direction, but their sum has no dynamical meaning because they are not applied to the same object. To decide whether a body accelerates, one must consider only the forces acting on that body.

Forces on the cart. Acting on the cart are: the force $\vv{F}_{c \to C}$ with which the horse pulls it forward (via the harness) and the friction force $\vv{F}_{a}$ from the ground on the wheels, directed backward. The horizontal resultant on the cart is

$F_{c \to C} - F_a = m_C\,a$

If the horse’s traction exceeds the friction, this resultant is positive and the cart accelerates. The cart’s reaction on the horse does not appear here, because it acts on the horse, not on the cart.

Forces on the horse. Acting on the horse are: the reaction $\vv{F}_{C \to c}$ from the cart, directed backward (this is the “reaction” the horse refers to), and — crucially — the force with which the ground pushes the hooves forward. The horse, by pushing the ground backward with its legs, receives from the ground (third law applied to the hoof-ground pair) a forward push. If this push exceeds the cart’s reaction, the horse too accelerates forward.

Conclusion. The two forces cited by the horse, $\vv{F}_{c \to C}$ and $\vv{F}_{C \to c}$, are indeed equal and opposite, but one acts on the cart and one on the horse: they do not cancel each other out. The horse-and-cart system moves thanks to the external force supplied by the ground. It is enough for the horse to push the ground backward for the whole system to accelerate forward.

$\ev{\text{Action and reaction act on different bodies: they do not cancel out. The cart moves.}}$

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Topics: Dynamics Concepts: Newton’s third law