When applying the impulse equation, be sure to calculate the vector change in velocity—for example, consider a mass of 10 kg acted on by a force that changes its velocity from −8 m/s to 3 m/s . This force imparted an impulse of (10 kg)[3 − (−8) m/s] = (10 kg)(11 m/s) = 110 N‐s.
An extremely important fundamental principle in physics is the law of conservation of momentum. The law states that if there is no external force acting on a system, the total momentum remains a constant, which provides a powerful way to analyze interactions between systems of objects. For example, if a rolling ball on a frictionless surface collides with another ball, the total momentum before and after the collision is the same. An interaction, therefore, can be examined without knowing the forces involved and the length of interaction time, which might be difficult to measure.
First, consider a head‐on collision (so it is not necessary to utilize two‐dimensional vectors, that is, consider only straight line motion). Imagine a mass ( m 1) with velocity ( v 1) hitting a mass ( m 2) that is initially at rest. The momentums are the following: before the collision, m 1 v 1, and after the collision, m 1 v 1 + m 2 v 2, where the primes indicate velocities after the interaction. From the law of conservation of momentum, the two expressions may be set equal to each other. Consider the special case where the two masses are equal on frictionless surface and stick together after the collision (have the same primed velocity). Then, total momentum before the collision equals total momentum after the collision, mv 1 = mv′ + mv′; therefore, v′ = (1/2)(v 1), or the final velocity is one‐half the original velocity because the effective mass has doubled.
Another way to state the law of conservation of momentum is that the change in momentum of the two objects must be equal and opposite. For example, two ice skaters are at rest in the center of frictionless ice (possible at least in the imagination). Let one have a relatively small mass (m) and the other a larger mass (M). Because they begin at rest, the initial momentum is zero. They then push apart in opposite directions. The total momentum must remain zero.
According to the law of conservation of momentum, Δp m = −Δp m or mv′ − 0 = −( MV′−0); therefore, if the large mass (M) is three times the smaller mass (m), v′ = −3V′, where v′ is the velocity of the small mass after the collision and V′ is the velocity of the large mass after the collision. The negative sign indicates velocities in opposite directions.
This same analysis holds for a person standing on frictionless ice who throws an object; it even holds for a rocket going to the moon. The ice skater throwing a glove attains equal momentum in the direction opposite to that of the thrown object. This basic principle is the same for a rocket accelerating in space. Spacecrafts utilize the law of conservation of momentum in getting an additional push from discharged rocket stages as well as from fuel. In particular, the Apollo space capsule returning from the moon was only a small percent of the total mass initially sent upward from the launch pad; therefore, acceleration of a rocket can be caused by either a change in velocity, by a change in its mass, or by changes in both velocity and mass (see Figure ). Thus, the expression of Newton's second law of motion, stated in terms of the change of momentum, is broader than the expression given only in terms of mass and acceleration.
A rocket ship gains momentum.
If two objects strike with a glancing blow, the motion will be two‐dimensional. For example, one ball ( m 1) with an initial velocity hits a second ball ( m 2), which is initially at rest. Figures and depict this situation with the first ball initially moving up from the bottom of the page. For the sake of simplicity, allow the two masses to be equal.
Two balls striking a glancing blow.
(a) Total momentum is conserved. (b) Equal and opposite momentum changes of the two balls.
The momentum vectors can be added to show the law of conservation of momentum. The vector addition in Figure (a) shows that the total of the two momentum vectors, p1′ and p2′, after the collision are equal to the total momentum before the collision. (Because only m 1 was moving, there was only one initial momentum vector, p1.) Figure (b) also shows the alternate method of using the law of conservation of momentum, that the change (difference) in momentum of m 1 is equal and opposite to that of m 2.
If the two masses are not equal, the velocity vectors must be adjusted so that the vectors represent momentum. For example, if one mass is three times the other, the velocity vectors of the larger mass must be lengthened by a factor of three before using the law of conservation of momentum.