The Horse-and-Cart Puzzle & Inertia
If the cart pulls back as hard as the horse pulls forward, how does anything move?
Lesson at a glanceShow the key points
- Action–reaction forces act on different bodies, so they cannot cancel in the equation for any one body.
- The horse moves forward because the ground pushes it forward harder than the cart pulls it back.
- The cart accelerates if the horse's pull exceeds the backward forces on the cart from the ground.
- Only forces ON the chosen system enter F = ma, and all of them must be included.
- Inertia is a body's resistance to changing its velocity; larger mass means smaller acceleration for the same force.
A horse pulls a cart forward. By the third law the cart pulls the horse back with an equal force. So the forces cancel and nothing should move. Something is wrong with that argument. What?
The paradox
Call the horse’s pull on the cart (forward) and the cart’s pull on the horse (backward). The third law guarantees . Their sum is zero. Why does the cart accelerate forward at all?
The flaw is that we added forces acting on two different bodies. The recipe says: choose a system first, then list the forces on it, all of them and only them.
Take the cart as the system
On the cart act the horse’s forward pull and a backward force from the road (friction and rolling resistance at the wheels). acts on the horse, so it has no business here. The cart accelerates forward if , with .
Take the horse as the system
On the horse act the cart’s backward pull and, crucially, the force of the road on its hooves. The horse pushes back on the road; the road pushes forward on the horse with a force (static friction). The horse accelerates forward if .
The forces and adjust themselves so that the horse and cart end up with the same acceleration and move together, as they must while the harness stays taut.
A 400 kg horse pulls a 300 kg cart. The ground pushes the horse forward with 2100 N, and resists the cart with 700 N. Find the acceleration and the force in the harness.
Inertia
In an inertial frame a body accelerates if and only if a net force acts on it. Loosely, a body does not change its state of rest or of uniform straight-line motion unless something forces it to. We call this reluctance to change velocity inertia.
- 1Forces on different bodies can never be added to decide one body's motion.
- 2Pick the system, include every force on it, and only those, then apply F = ma.
- 3Horse and cart move because the ground pushes the horse forward harder than it resists the cart.
- 4Inertia is resistance to acceleration; mass measures it.
Check yourself
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