# Newton's laws of motion

Canonical: https://duckyhelper.com/learn/physics/newtons-laws-of-motion/
Updated: 2026-10-01

Newton's three laws explain how forces change motion. First law: an object keeps the same velocity, staying still or moving straight at a steady speed, unless a net force acts on it. Second law: net force equals mass times acceleration, \(\Sigma F = ma\). Third law: when object A pushes on object B, B pushes back on A with a force of equal size in the opposite direction.

## Key ideas

### First law: inertia

Objects resist changes to their motion. This resistance is called **inertia**, and more mass means more inertia. A hockey puck slides a long way on ice because almost no force acts to stop it. If the net force is zero, the acceleration is zero: the object is at rest or moving at constant velocity.

### Second law: net force and acceleration

$$
\Sigma F = m a
$$

- \(\Sigma F\) is the net force, the sum of all forces on the object with directions included, in newtons (N).
- \(m\) is the mass, in kilograms (kg).
- \(a\) is the acceleration, in m/s². It points the same way as the net force.
- One newton is the force that accelerates 1 kg at 1 m/s²: \(1\ \text{N} = 1\ \text{kg·m/s}^2\).

**Weight** is the force of gravity on an object. It is a force, so it is in newtons, while mass is in kilograms:

$$
W = m g, \qquad g = 9.8\ \text{m/s}^2 \text{ on Earth}
$$

### Third law: forces come in pairs

When you push on a wall, the wall pushes back on you just as hard. The two forces in a pair are equal in size, opposite in direction, the same type of force, and act on **different** objects. That last part is why they never cancel each other out.

**The three laws at a glance**

| Law | In one line | Everyday example |
| --- | --- | --- |
| First | No net force means no change in velocity | You lurch forward when a bus brakes, because your body keeps moving |
| Second | \(\Sigma F = ma\) | An empty cart is easier to speed up than a full one |
| Third | Forces come in equal and opposite pairs | A swimmer pushes water back, and the water pushes the swimmer forward |

## Worked examples

**Example 1: force from mass and acceleration**

Problem: A skateboarder and board have a total mass of 55 kg and speed up at 1.8 m/s². What net force acts on them?

1. Use the second law.

   $$
   \Sigma F = m a = (55\ \text{kg})(1.8\ \text{m/s}^2) = 99\ \text{N}
   $$
2. The net force points the same way as the acceleration: forward.

Answer: 99 N forward

**Example 2: two forces in opposite directions**

Problem: A 20.0 kg crate is pushed right with 150 N while friction pulls left with 90 N. What is its acceleration?

1. Take right as positive and add the forces with their signs.

   $$
   \Sigma F = 150\ \text{N} - 90\ \text{N} = 60\ \text{N}
   $$
2. Divide by mass.

   $$
   a = \frac{\Sigma F}{m} = \frac{60\ \text{N}}{20.0\ \text{kg}} = 3.0\ \text{m/s}^2
   $$

Answer: 3.0 m/s² to the right

**Example 3: mass vs weight**

Problem: A student has a mass of 60.0 kg. What is the student's weight on Earth, and on the Moon, where \(g = 1.62\ \text{m/s}^2\)?

1. Weight on Earth.

   $$
   W = mg = (60.0\ \text{kg})(9.8\ \text{m/s}^2) = 588\ \text{N}
   $$
2. Weight on the Moon.

   $$
   W = (60.0\ \text{kg})(1.62\ \text{m/s}^2) = 97.2\ \text{N}
   $$
3. The mass is 60.0 kg in both places. Only the pull of gravity changes.

Answer: 588 N on Earth and 97.2 N on the Moon

**Example 4: an elevator speeding up**

Problem: A 50.0 kg person stands on a scale in an elevator that accelerates upward at 2.0 m/s². What does the scale read?

1. Two forces act on the person: the scale's normal force \(F_N\) up and weight \(mg\) down. Up is positive.

   $$
   F_N - mg = ma
   $$
2. Solve for \(F_N\).

   $$
   F_N = m(g + a) = (50.0\ \text{kg})(9.8 + 2.0)\ \text{m/s}^2 = 590\ \text{N}
   $$
3. The scale shows the normal force, which is more than the person's weight of 490 N while the elevator speeds up.

Answer: 590 N

## Common mistakes and how to fix them

- **Thinking motion needs a force.** A moving object needs no force to keep moving. Forces change motion. Fix: if velocity is constant, the net force is zero.
- **Using one force instead of the net force.** Fix: \(\Sigma F\) means add every force with its direction, then use \(ma\).
- **Mixing up mass and weight.** Mass (kg) is how much matter. Weight (N) is the gravity force. Fix: \(W = mg\).
- **Cancelling third law pairs.** The pair acts on two different objects, so they never cancel on one object. Fix: when you add forces, include only forces acting on your chosen object.

**Practice problems**

1. A net force of 12 N acts on a 4.0 kg cart. What is its acceleration?
   A. 3.0 m/s²
   B. 48 m/s²
   C. 0.33 m/s²
   D. 16 m/s²

   Answer: 3.0 m/s². \(a = \Sigma F / m = 12 \div 4.0 = 3.0\) m/s².

2. When you walk, your foot pushes backward on the ground. What is the third law partner of that force?
   A. The ground pushes forward on your foot
   B. Gravity pulls you down
   C. Friction pushes backward on your foot
   D. Your other foot pushes forward

   Answer: The ground pushes forward on your foot. The pair has the same two objects swapped (foot on ground, ground on foot), equal size and opposite direction. That forward push from the ground is what moves you.

3. A car drives at a constant 25 m/s on a straight, flat highway. What is the net force on it?
   A. 0 N
   B. Equal to its weight
   C. Forward, to keep it moving
   D. It depends on its speed

   Answer: 0 N. Constant velocity means zero acceleration, so by \(\Sigma F = ma\) the net force is zero. The engine's push just balances friction and air drag.

4. An object weighs 196 N on Earth. What is its mass?

   Answer: 20.0 kg. \(m = W/g = 196 \div 9.8 = 20.0\) kg.

5. The same 10. N net force acts on a 2.0 kg ball and on a 5.0 kg ball. How many times larger is the small ball's acceleration?

   Answer: 2.5 times. \(a = F/m\): 5.0 m/s² for the 2.0 kg ball and 2.0 m/s² for the 5.0 kg ball. \(5.0 \div 2.0 = 2.5\).

## Frequently asked questions

### If action and reaction are equal, how does anything move?

The two forces act on different objects. When a horse pulls a cart, the cart pulls back on the horse, but the horse also pushes on the ground and the ground pushes it forward. To know if an object accelerates, add only the forces acting on that one object.

### Is inertia a force?

No. Inertia is a property of matter, its resistance to changes in motion, and it depends only on mass. When you feel pushed forward as a car brakes, no force pushes you. Your body just keeps moving while the car slows.

### What is the normal force?

It is the push a surface gives to an object resting on it, at a right angle to the surface. On a flat floor with nothing else pushing, it equals the object's weight. On a ramp or in an accelerating elevator it is different, so always find it from \(\Sigma F = ma\).

## Sources

- [OpenStax College Physics 2e, 4.2 Newton's First Law of Motion: Inertia](https://openstax.org/books/college-physics-2e/pages/4-2-newtons-first-law-of-motion-inertia), accessed 2026-10-01
- [OpenStax College Physics 2e, 4.3 Newton's Second Law of Motion: Concept of a System](https://openstax.org/books/college-physics-2e/pages/4-3-newtons-second-law-of-motion-concept-of-a-system), accessed 2026-10-01
- [OpenStax College Physics 2e, 4.4 Newton's Third Law of Motion: Symmetry in Forces](https://openstax.org/books/college-physics-2e/pages/4-4-newtons-third-law-of-motion-symmetry-in-forces), accessed 2026-10-01

## Related

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## Try asking Ducky

- "Check my free body diagram for the elevator problem. Did I miss a force?"
- "I think the normal force and weight are a third law pair. Am I right?"
- "Give me three net force problems that get harder each time."

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