Newton's three laws explain why objects move, speed up, slow down or stay put. They apply to a soccer ball, a car, a skater and a spacecraft alike. Together they connect a single idea, the force, with what happens to the motion of a body.
Each law is short to state but easy to misread. This article gives the statement, an everyday example and the math of each law, with two simulators: a force lab for the first two laws and a pair of skaters for the third.
First law: inertia
An object keeps its state of motion, at rest or moving in a straight line at constant speed, unless a net force acts on it. This tendency to resist changes in motion is called inertia, and the mass of the object measures it.
- A passenger lurches forward when a car brakes: the body keeps moving while the car slows down. A seat belt provides the force that stops it.
- A hockey puck on smooth ice slides a long way because there is almost no friction to slow it.
- A tablecloth can be yanked out from under dishes because the dishes stay in place for the short time the force acts on them.
Notice what the law says about forces: no force is needed to keep an object moving at constant speed. Forces are needed to change the speed or the direction. A moving block eventually stops on a table only because friction pushes against it.
Second law: F = ma
When a net force acts on an object, it accelerates in the direction of that force. The acceleration is proportional to the net force and inversely proportional to the mass:
Second law
$latex F_{net} = m \times a$
The net force $latex F_{net}$ is the vector sum of all forces on the object (see adding vectors). The unit of force is the newton: 1 N is the force that gives 1 kg an acceleration of 1 m/s², so $latex 1\ \text{N} = 1\ \text{kg} \cdot \text{m/s}^2$. Solving for the acceleration, $latex a = \dfrac{F_{net}}{m}$: push twice as hard and the acceleration doubles, load the cart with twice the mass and it halves.
Mass and weight
Mass is the amount of matter and does not change from place to place. Weight is the gravitational force on that mass, $latex W = m \times g$. It depends on where you are: the acceleration due to gravity is 9.8 m/s² on Earth but much less on the Moon.
| Body | g (m/s²) | Weight of a 70 kg person |
|---|---|---|
| Earth | 9.8 | 686 N |
| Mars | 3.71 | 259.7 N |
| Moon | 1.62 | 113.4 N |
Force lab: push a block
Set a push force, a mass and a friction coefficient, then press Play. You push during the first 5 seconds and let go afterwards. With no friction the block keeps its speed after you let go (first law). With friction it slows down. Compare the speed plot for different masses to see the second law at work.
Force lab: push a block and watch what happens
Third law: action and reaction
When body A pushes on body B, body B pushes back on A with a force of the same size and opposite direction. The two forces always act on different bodies, so they never cancel each other.
Third law
$latex \vec{F}_{A\ on\ B} = -\vec{F}_{B\ on\ A}$
- When you walk, your foot pushes the ground backward and the ground pushes you forward. That forward push is what moves you.
- A rocket pushes hot gas downward and the gas pushes the rocket upward. It works in space, where there is nothing to push against but the gas itself.
- A swimmer pushes water backward and is pushed forward.
The skaters below push each other with equal and opposite forces for half a second. The forces match, but the accelerations do not unless the masses are the same, because $latex a = F / m$.
Third law: two skaters push apart
Solved examples
Example 1
A 10 kg box is pushed by a 50 N force on a frictionless floor. Find its acceleration, and its speed and distance after 4 s starting from rest.
$$a = \frac{F}{m} = \frac{50}{10} = 5\ \dfrac{\text{m}}{\text{s}^2}$$
$$v = a t = 5 \times 4 = 20\ \text{m/s}$$
$$d = \frac{1}{2} a t^2 = \frac{1}{2} \times 5 \times 4^2 = 40\ \text{m}$$
The acceleration is 5 m/s². After 4 s the box moves at 20 m/s and has covered 40 m. The kinematic formulas come from the equations for constant acceleration.
Example 2
The same box is pushed with 50 N, but now the floor has a friction coefficient of 0.2. Find the acceleration.
$$f = \mu m g = 0.2 \times 10 \times 9.8 = 19.6\ \text{N}$$
$$F_{net} = 50 - 19.6 = 30.4\ \text{N}$$
$$a = \frac{30.4}{10} = 3.04\ \dfrac{\text{m}}{\text{s}^2}$$
Friction takes away part of the push, so the acceleration drops to 3.04 m/s². This is the setup shown in the force lab.
Example 3
Find the weight of a 70 kg person on Earth and on the Moon ($latex g = 1.62\ \text{m/s}^2$).
$$W_{Earth} = 70 \times 9.8 = 686\ \text{N}$$
$$W_{Moon} = 70 \times 1.62 = 113.4\ \text{N}$$
The weight is 686 N on Earth and 113.4 N on the Moon. The mass is 70 kg in both places.
Example 4
Skater A (50 kg) pushes skater B (75 kg) with a force of 150 N on frictionless ice. Find each acceleration.
By the third law, B pushes A with 150 N in the opposite direction. Apply the second law to each skater:
$$a_A = \frac{150}{50} = 3\ \dfrac{\text{m}}{\text{s}^2}$$
$$a_B = \frac{150}{75} = 2\ \dfrac{\text{m}}{\text{s}^2}$$
The lighter skater accelerates at 3 m/s² and the heavier one at 2 m/s², in opposite directions. Equal forces, unequal accelerations.
Common mistakes
- Thinking a force is needed to keep moving. A constant velocity means the net force is zero. A moving object that slows down is being acted on by friction or drag.
- Mixing up mass and weight. Mass is in kilograms and is the same everywhere. Weight is a force in newtons and depends on g.
- Using one force instead of the net force. In $latex F = ma$, F is the sum of all forces. Subtract friction before dividing by the mass.
- Canceling action and reaction. They act on different bodies, so they never appear together in the same free-body diagram.
- Forgetting vector directions. Forces pointing in opposite directions subtract. Choose a positive direction and keep the signs.
Practice problems
Problem 1
A 1200 kg car is acted on by a net force of 3000 N. What is its acceleration?
Answer
$$a = \frac{3000}{1200} = 2.5\ \dfrac{\text{m}}{\text{s}^2}$$
The acceleration is 2.5 m/s².
Problem 2
What net force gives a 0.5 kg ball an acceleration of 12 m/s²?
Answer
$$F = m a = 0.5 \times 12 = 6\ \text{N}$$
The net force is 6 N.
Problem 3
A 60 kg astronaut visits Mars, where $latex g = 3.71\ \text{m/s}^2$. Find the weight on Earth and on Mars.
Answer
$$W_{Earth} = 60 \times 9.8 = 588\ \text{N}$$
$$W_{Mars} = 60 \times 3.71 = 222.6\ \text{N}$$
The weight is 588 N on Earth and 222.6 N on Mars.
Problem 4
A 20 kg crate is pushed with 100 N on a floor with friction coefficient 0.3. Find its acceleration.
Answer
$$f = 0.3 \times 20 \times 9.8 = 58.8\ \text{N}$$
$$a = \frac{100 - 58.8}{20} = 2.06\ \dfrac{\text{m}}{\text{s}^2}$$
The acceleration is 2.06 m/s².
Problem 5
An 80 kg person stands on a scale in an elevator that accelerates upward at 1.5 m/s². What does the scale read, in newtons?
Answer
Two forces act on the person: the normal force $latex N$ upward and the weight downward. Taking up as positive, $latex N - m g = m a$.
$$N = m (g + a) = 80 \times (9.8 + 1.5) = 904\ \text{N}$$
The scale reads 904 N, more than the resting weight of 784 N.
Problem 6
A 60 kg skater pushes a 40 kg friend with 120 N on frictionless ice. Find both accelerations.
Answer
$$a_{60} = \frac{120}{60} = 2\ \dfrac{\text{m}}{\text{s}^2}$$
$$a_{40} = \frac{120}{40} = 3\ \dfrac{\text{m}}{\text{s}^2}$$
The 60 kg skater accelerates at 2 m/s² and the 40 kg friend at 3 m/s², in opposite directions.
Frequently asked questions
Is the first law just a special case of the second?
Mathematically yes: if the net force is zero, the acceleration is zero. But the first law also defines the frames of reference in which the second law works, which is why it is stated separately.
Do action and reaction cancel out?
No. They act on different bodies. To know whether a body accelerates, you add only the forces that act on that body.
Why do we need friction to walk?
To move forward, your foot must push the ground backward. The ground pushes you forward with the same force, and without friction your foot would slip, as on ice.
What is the difference between mass and weight?
Mass measures how much matter an object has and how strongly it resists changes in motion. Weight is the force of gravity on it. A person has the same mass on the Moon, but weighs about one sixth as much.
Related calculators
- Vector calculator: add and resolve forces as vectors.
- kg to lb calculator: convert masses between kilograms and pounds before computing weights.
- Speed, distance and time calculator: connect the acceleration you find with the motion.
See also
- Acceleration due to gravity: the value of g and why it changes.
- Kinematic equations for constant acceleration: from acceleration to speed and distance.
- Velocity vs time graph: the slope of the graph is the acceleration.
- Adding vectors: how to find the net force.
- Components of a vector: split forces along the axes.