If forces are equal and opposite, how does a horse pull a cart?

The horse's pull on the cart and the cart's pull on the horse are indeed equal and opposite. However, the horse also pushes backward on the ground, and the ground pushes forward on the horse. As long as the forward force from the ground on the horse is greater than the backward pull from the cart, the horse will accelerate forward.

Does a heavier object push back harder than a lighter object?

No. The forces are always exactly equal. If a massive truck collides with a tiny bicycle, the force on the truck is exactly the same as the force on the bicycle. The bicycle just gets damaged more because it has much less mass, so that same amount of force causes a much larger and more destructive acceleration.

Do the action and reaction forces happen at the same time?

Yes, they occur simultaneously. The moment you push on an object, it is pushing back on you. There is no delay. Because they happen at the exact same time, it doesn't actually matter which one you call the 'action' and which you call the 'reaction'.

What is Newton's third law really saying?

Newton's third law is often quoted as 'for every action, there is an equal and opposite reaction.' But what it is really saying is that forces never happen in isolation. A force is simply an interaction between two objects. You cannot touch something without it touching you back.

When Object A pushes or pulls on Object B, Object B pushes or pulls back on Object A with the exact same amount of force, but in the exact opposite direction. This happens instantly and applies to every force in the universe, whether it is gravity pulling an apple down or a rocket pushing exhaust gas out.

The action-reaction pair

Think of forces like a handshake. You can't shake someone's hand without them shaking yours. The 'action' and 'reaction' are just two sides of the exact same interaction. If you press your finger against a wall, the wall pushes back against your finger. You can feel the wall's force squishing the tip of your finger. The harder you push the wall, the harder the wall pushes back.

In physics terms, if object A exerts a force on object B, then object B exerts an equal and opposite force on object A. We can write this mathematically as $F_{A \to B} = -F_{B \to A}$ . The negative sign just means the direction is opposite.

Why it works

Nature is fundamentally balanced when it comes to interactions. A force isn't something an object 'has'; it is something an object 'does' to another object. Because a force is a mutual interaction, it must be felt by both participants. The universe doesn't allow one-way forces. If Earth's gravity pulls down on you with 600 Newtons of force, your body's gravity is pulling up on the Earth with exactly 600 Newtons of force. The interaction is completely symmetrical.

Where students slip up

The most common mistake is thinking that because the forces are equal and opposite, they should cancel each other out, meaning nothing would ever move. The trick is to remember that these two forces act on different objects.

If you push a shopping cart, your pushing force acts on the cart. The cart's reaction force acts on you. Because the forces act on two completely different things, they do not cancel each other out. To find out if the cart moves, you only look at the forces acting on the cart, not the forces the cart exerts on other things.

Worked through

A 50 kg ice skater faces a 75 kg ice skater on a frictionless ice rink. The 50 kg skater pushes the 75 kg skater with a force of 150 N to the right. What is the magnitude and direction of the force exerted on the 50 kg skater? What are the resulting accelerations of both skaters?

According to Newton's third law, the force exerted by the 50 kg skater on the 75 kg skater is equal and opposite to the force exerted by the 75 kg skater on the 50 kg skater.

Force on the 50 kg skater: 150 N to the left.

Now, we use Newton's second law ( $a = \frac{F}{m}$ ) to find their accelerations. Even though the forces are the same, their accelerations will be different because their masses are different.

Acceleration of the 75 kg skater: $a = \frac{150}{75} = 2 \text{ m/s}^2$ to the right.

Acceleration of the 50 kg skater: $a = \frac{150}{50} = 3 \text{ m/s}^2$ to the left.

The forces are exactly equal, but the lighter skater accelerates more.

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Where this comes from: OpenStax University Physics Volume 1, Chapter 5: Newton's Laws of Motion · Khan Academy, Unit: Forces and Newton's laws of motion