Lesson 2: The Law of Momentum Conservation

The Law of Action-Reaction (Revisited)

A collision is an interaction between two objects which have made contact (usually) with each other. As in any interaction, a collision results in a force being applied to the two colliding objects. Such collisions are governed by Newton's laws of motion. In the second unit of The Physics Classroom, Newton's third law of motion was introduced and discussed. It was said that...

... in every interaction, there is a pair of forces acting on the two interacting objects. The size of the force on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.

Newton's third law of motion is naturally applied to collisions between two objects. In a collision between two objects, both objects experience forces which are equal in magnitude and opposite in direction. Such forces cause one object to speed up (gain momentum) and the other object to slow down (lose momentum). According to Newton's third law, the forces on the two objects are equal in magnitude. While the forces are equal in magnitude and opposite in direction, the acceleration of the objects are not necessarily equal in magnitude. In accord with Newton's second law of motion, the acceleration of an object is dependent upon both force and mass. Thus, if the colliding objects have unequal mass, they will have unequal accelerations as a result of the contact force which results during the collision.

Consider the collision between the club head and the golf ball in the sport of golf. When the club head of a moving golf club collides with a golf ball at rest upon a tee, the force experienced by the club head is equal to the force experienced by the golf ball. Most observers of this collision have difficulty with this concept because they perceive the high speed given to the ball as the result of the collision. They are not observing unequal forces upon the ball and club head, but rather unequal accelerations. Both club head and ball experience equal forces, yet the ball experiences a greater acceleration due to its smaller mass. In a collision, there is a force on both objects which causes an acceleration of both objects; the forces are equal in magnitude and opposite in direction, yet the least massive object receives the greatest acceleration.

Consider the collision between a moving seven-ball and an eight-ball that is at rest in the sport of billiards. When the seven-ball collides with the eight-ball, each ball experiences an equal force directed in opposite directions. The rightward moving seven-ball experiences a leftward force which causes it to slow down; the eight-ball experiences a rightward force which causes it to speed up. Since the two balls have equal masses, they will also experience equal accelerations. In a collision, there is a force on both objects which causes an acceleration of both objects; the forces are equal in magnitude and opposite in direction. For collisions between equal-mass objects, each object experiences the same acceleration.

Consider the interaction between a male and female figure skater in pair figure skating. A woman (m = 45 kg) is kneeling on the shoulders of a man (m = 70 kg); the pair is moving along the ice at 1.5 m/s. The man gracefully tosses the woman forward through the air and onto the ice. The woman receives the forward force and the man receives a backward force. The force on the man is equal in magnitude and opposite in direction to the force on the woman. Yet the acceleration of the woman is greater than the acceleration of the man due to the smaller mass of the woman.

Many observers of this interaction have difficulty believing that the man experienced a backward force. "After all," they might argue, "the man did not move backward." Such observers are presuming that forces cause motion; that is a backward force would cause a backward motion. This is a common misconception that has been addressed elsewhere in The Physics Classroom. Forces cause acceleration, not motion. The male figure skater experiences a backwards (you might say "negative") force which causes his backwards (or "negative") acceleration; that is, the man slowed down while the woman sped up. In every interaction (with no exception), there are forces acting upon the two interacting objects which are equal in magnitude and opposite in direction.

Collisions are governed by Newton's laws. The law of action-reaction (Newton's third law) explains the nature of the forces between the two interacting objects. According to the law, the force exerted by object 1 upon object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 upon object 1.

Check Your Understanding

Express your understanding of Newton's third law by answering the following questions. Depress the mouse on the "pop-up" menu to view the answers.

1. While driving down the road, Anna Litical observed a bug striking the windshield of her car. Quite obviously, a case of Newton's third law of motion. The bug hit the windshield and the windshield hit the bug. Which of the two forces is greater: the force on the bug or the force on the windshield?

Depress mouse to see answer. Trick Question! Each force is the same size. For every action, there is an equal ... (equal!). The fact that the bug splatters only means that with its smaller mass, it is less able to withstand the larger acceleration resulting from the interaction.

2. Rockets are unable to accelerate in space because ...

there is no air in space for the rockets to push off of.

there is no gravity is in space.

there is no air resistance in space.

...nonsense! Rockets do accelerate in space.

Depress mouse to see answer. The answer is D. It is a common misconception that rockets are unable to accelerate in space. The fact is that rockets do accelerate. Rockets are able to accelerate due to the fact that they burn fuel and push the exhaust gases in a direction opposite the direction which they wish to accelerate.

3. A gun recoils when it is fired. The recoil is the result of action-reaction force pairs. As the gases from the gunpowder explosion expand, the gun pushes the bullet forwards and the bullet pushes the gun backwards. The acceleration of the recoiling gun is ...

greater than the acceleration of the bullet.

smaller than the acceleration of the bullet.

the same size as the acceleration of the bullet.

Depress mouse to see answer. The answer is B. The force on the gun equals the force on the bullet. Yet, acceleration depends on both force and mass. The bullet has a greater acceleration due to the fact that it has a smaller mass. Remember: acceleration and mass are inversely proportional.

4. Why is it important that an airplane wing be designed so that it deflects oncoming air downward?

Depress mouse to see answer. This can be explained by Newton's third law of motion. The more air that the wing can push down, the more that the air is able to push the wing up. If enough air is pushed downward, then the reaction to this will result in sufficient upward push on the wing and the plane to provide the lift necessary to elevate the plane off the ground.

5. Would it be a good idea to jump from a rowboat to a dock that seems within jumping distance? Explain.

Depress mouse to see answer. No! Don't do this at home (at least, not if you wish to dock the boat)! As you jump to reach the dock, the rowboat pushes you forward (the action) and thus you push the rowboat backwards. You will indeed reach the dock; your rowboat will be several feet away.

6. If we throw a ball horizontally while standing on roller skates, we roll backward with a momentum that matches that of the ball. Will we roll backward if we go through the motion of throwing the ball without letting go of it? Explain.

Depress mouse to see answer. The overall motion of a person who merely goes through the motion of throwing the ball (without letting go) will be "null." Such a person will roll backwards and then forwards. Yet when finished, the person will finsih where she started. Recall the demonstration of this in class.

7. Suppose there are three astronauts outside a spaceship and two of them decide to play catch with the other woman. All three astronauts weigh the same on Earth and are equally strong. The first astronaut throws the second astronaut towards the third astronaut and the game begins. Describe the motion of these women as the game proceeds. Assume each toss results from the same-sized "push." How long will the game last?

Depress mouse to see answer. The game will last two throws and one catch. When astronaut #1 throws astronaut #2, the two astronauts will travel opposite directions at the same speed (action-reaction). When astronaut #3 catches astronaut #2, astronaut #2 will slow down to half her speed and move together with astronaut #3. Now astronaut #1 is moving leftward with the original speed and astronaut #2 and #3 are moving rightward with half the original speed. When astronaut #3 pushes #2, the greatest speed which #2 can half is half the original speed in the opposite direction. The game is now over for astronaut #2 can never catch up with astronaut #1.

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