Conservation Laws

A rocket ship taking off, the recoil of a rifle, and a bank-shot in a pool are examples which demonstrate the conservation of linear momentum. Linear momentum is defined as the product of an object's mass and its velocity. For example, the linear momentum of a 220 lb (100 kg) football-linebacker traveling at a speed of 10 MPH (16 km/h) is exactly the same as the momentum of a 110 lb (50 kg) sprinter traveling at 20 MPH (32 km/h). Since the velocity is both the speed and direction of an object, the linear momentum is also specified by a certain direction.

The linear momentum of one or more objects is conserved when there are no external forces acting on those objects. For example, consider a rocket-ship in deep outer space where the force of gravity is negligible. Linear momentum will be conserved since the external force of gravity is absent. If the rocket-ship is initially at rest, its momentum is zero since its speed is zero (Figure 1a). If the rocket engines are suddenly fired, the rocket-ship will be propelled forward (Figure 1b). For linear momentum to be conserved, the final momentum must be equal to the initial momentum, which is zero. Linear momentum is conserved if one takes into account the burnt fuel that is ejected out the back of the rocket. The positive momentum of the rocket-ship going forward is equal to the negative momentum of the fuel going backward (note that the direction of motion is used to define positive and negative). Adding these two quantities yields Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group. zero. It is important to realize that the rocket's propulsion is not achieved by the fuel pushing on anything. In outer space there is nothing to push on! Propulsion is achieved by the conservation of linear momentum. An easy way to demonstrate this type of propulsion is by propelling yourself on a frozen pond. Since there is little friction between your ice skates and the ice, linear momentum is conserved. Throwing an object in one direction will cause you to travel in the opposite direction.

Even in cases where the external forces are significant, the concept of conservation of linear momentum can be applied to a limited extent. An instance would be the momentum of objects that are affected by the external force of gravity. For example, a bullet is fired at a clay pigeon that has been launched into the air. The linear momentum of the bullet and clay pigeon at the instant just before impact is equal to the linear momentum of the bullet and hundreds of shattered clay pieces at the instant just after impact. Linear momentum is conserved just before, during, and just after the collision (Figure 2). This is true is because the external force of gravity does not significantly affect the momentum of the objects within this narrow time period. Many seconds later, however, gravity will have had a significant influence and the total momentum of the objects will not be the same as just before the collision.

There are many examples that illustrate the conservation of linear momentum. When we walk down the road, our momentum traveling forward is equal to the momentum of the earth traveling backward. Of course, the mass of Earth is so large compared to us that its velocity will be negligible. (A simple calculation using the 220 lb [100 kg] linebacker shows that as he travels forward at 10 mph [16 kph]. Earth travels backward at a speed of 9 trillionths of an inch per century!) A better illustration is to walk forward in a row-boat and you will notice that the boat travels backward relative to the water. When a rifle is fired, the recoil you feel against your shoulder is due to the momentum of the rifle which is equal but in the opposite direction to the momentum of the bullet. Again, since the rifle is so much heavier than the bullet, its velocity will be correspondingly less than the bullet's. Conservation of linear momentum is the chief reason that heavier cars are safer than lighter cars. In a head-on collision with two cars traveling at the same speed, the motion of the two cars after the collision will be along the original direction of the larger car due to its larger momentum. Conservation of linear momentum is used to give space probes an extra boost when they pass planets. The momentum of the planet as it circles the sun in its orbit is given to the passing space probe, increasing its velocity on its way to the next planet. In all of the experiments ever attempted, there has been never been a violation of the law of conservation of linear momentum. This applies to all objects ranging in size from galaxies to subatomic particles.