Newton's third law
Press your palm against a wall and you can feel the wall pressing back. That is not a coincidence or a reaction that comes a moment later. Every force is one half of a pair, and this lesson is about those pairs.
By the end of this lesson you can
- state Newton's third law and identify action-reaction pairs;
- explain why the two forces in a pair never cancel each other;
- use the second and third laws together to compare the accelerations of two interacting objects.
Forces come in pairs
Think first
A car drives into a concrete barrier. The car obviously pushes on the barrier. Does the barrier push on the car? How do you know?
Compare your answer
Yes. The car stops very quickly, so it has a large backward acceleration, and something must push it backward: the barrier. The car pushes the barrier forward, and at the same time the barrier pushes the car backward.
A force is always an interaction between two objects. When your hand pushes a door, the door pushes your hand. When a rope pulls a bucket up, the bucket pulls down on the rope. Newton recognized that a single, isolated force cannot exist.
Newton's third law
Whenever A pushes or pulls on B, B pushes or pulls back on A with a force of the same size, aimed the opposite way:
$$\vec F_{\text{A on B}} = -\vec F_{\text{B on A}}$$The two forces are called an action-reaction pair. Which one you call the "action" does not matter. Both forces appear at exactly the same instant and disappear at exactly the same instant. The everyday meaning of "reaction", something that comes after, is misleading here.
Two skaters push apart
InteractiveScroll sideways to see the whole demo, or turn your phone sideways.
Try this
- Set both skaters to the same mass and push. Compare their speeds.
- Make skater 2 twice as heavy as skater 1. Are the force arrows still equal? Are the speeds?
- Use \(a = F/m\) to explain why the lighter skater always ends up faster.
Why the pair does not cancel
If the two forces are equal and opposite, why does anything ever move? Because the two forces act on different objects. Forces only cancel when they act on the same object.
Think of kicking a soccer ball. Your foot pushes the ball forward; the ball pushes your foot backward with the same force. To predict the ball's motion, draw the free-body diagram of the ball. Only the foot's push on the ball appears there, so the ball has a large net force and flies off. The ball's push on your foot belongs in your foot's free-body diagram, and you feel it.
Common mistake
A book rests on a table. Gravity pulls the book down and the table pushes the book up with an equal force. These two are not a third-law pair: both act on the book. They are equal because the book is in equilibrium (the first law). The real pairs are: Earth pulls the book / the book pulls Earth, and the table pushes the book / the book pushes the table.
Worked example 1
Two skaters stand face to face on smooth ice: Amira (50 kg) and Ben (75 kg). They push on each other's hands, and for a moment Amira pushes Ben with 150 N. Find the size of each skater's acceleration.
- Use the third law. If Amira pushes Ben with 150 N, Ben pushes Amira with 150 N in the opposite direction, at the same time.
- Apply the second law to Amira. The only horizontal force on her is Ben's push: $$a_{\text{Amira}} = \frac{150\ \text{N}}{50\ \text{kg}} = 3.0\ \text{m/s}^2$$
- Apply the second law to Ben. $$a_{\text{Ben}} = \frac{150\ \text{N}}{75\ \text{kg}} = 2.0\ \text{m/s}^2$$
Equal forces, unequal accelerations: 3.0 m/s² for Amira and 2.0 m/s² for Ben, in opposite directions.
Field forces come in pairs too
The third law is not only about touching. When you drop a ball, Earth pulls the ball down. At the same moment, the ball pulls Earth up with a force of the same size. So Earth accelerates toward the ball as the ball falls. You never notice, because Earth's mass is enormous.
Worked example 2
A 0.50 kg ball is falling. Earth pulls it down with 4.9 N. Earth's mass is \(6.0 \times 10^{24}\ \text{kg}\). What is Earth's acceleration toward the ball?
- Use the third law. The ball pulls Earth upward with 4.9 N.
- Apply the second law to Earth. $$a_{\text{Earth}} = \frac{4.9\ \text{N}}{6.0 \times 10^{24}\ \text{kg}} = 8.2 \times 10^{-25}\ \text{m/s}^2$$
About \(8 \times 10^{-25}\ \text{m/s}^2\): real, but far too small to ever measure.
Check yourself
Question 1
A swimmer pushes the water backward with her hands. What is the reaction force?
- The water pushes the swimmer forward.
- Gravity pulls the swimmer down.
- The water pushes the swimmer up (buoyancy).
- The swimmer's legs push the water backward.
Show solution
The pair partner of "hands push water backward" is "water pushes hands forward". Swap the two objects and reverse the direction.
Question 2
A small car collides head-on with a large truck. During the collision, how do the forces on the two vehicles compare, and which has the larger acceleration?
- The truck feels the larger force; the car has the larger acceleration.
- The car feels the larger force; the car has the larger acceleration.
- The forces are equal; the car has the larger acceleration.
- The forces are equal; the accelerations are equal.
Show solution
The third law makes the forces equal in size. The car has less mass, so by \(a = F/m\) its acceleration is larger. That is why people in the smaller vehicle are at greater risk.
Question 3
A child pulls a wagon with a rope. By the third law, the wagon pulls back on the child just as hard. So how can the wagon ever start moving?
Show solution
The two forces act on different objects. The wagon's motion depends only on forces on the wagon: the rope's pull forward and friction backward. If the pull is bigger than friction, the wagon accelerates. The wagon's pull on the child acts on the child (whose feet push against the ground to stay in place).
Question 4
Why must a climber pull down on a rope in order to move up?
Show solution
When the climber pulls down on the rope, the rope pulls up on the climber with an equal force. If that upward pull is larger than the climber's weight, the net force on the climber is upward and they accelerate up.
Question 5
An astronaut floating in space (90 kg including her suit) throws a 4.0 kg tool. During the throw she pushes on the tool with 27 N. What is the size of her acceleration during the throw?
Show solution
By the third law the tool pushes back on her with 27 N, so \(a = \dfrac{27\ \text{N}}{90\ \text{kg}} = 0.30\ \text{m/s}^2\), away from the tool. For comparison, the tool's acceleration is \(\dfrac{27\ \text{N}}{4.0\ \text{kg}} \approx 6.8\ \text{m/s}^2\).
Summary
- Every force is part of an interaction between two objects; single forces do not exist.
- Newton's third law: \(\vec F_{\text{A on B}} = -\vec F_{\text{B on A}}\), at the same instant.
- The partners in a pair push on two separate objects, so they cannot cancel.
- Equal forces can produce very different accelerations: the object with less mass accelerates more.
Finished?
Mark the lesson as complete to keep track of your progress. Your progress is saved in this browser only.