Sunday, October 26, 2014

Momentum: Linear Momentum, Impulse, Conservation of Linear Momentum and Collisions

Momentum

In This Chapter:

Linear Momentum

Impulse

Conservation of Linear Momentum

Collisions

Linear Momentum

Work and energy are scalar quantities that have no directions associated with them. When two or more bodies interact with one another, or a single body breaks up into two or more new bodies, the various directions of motion cannot be related by energy considerations alone. The vector quantities, called linear momentum and impulse, are important in analyzing such events.

The linear momentum (usually called simply momentum) of a body of mass m and velocity v is the product of m and v:

Momentum = mv

The units of momentum are kilogram-meters per second and slug-feet per second. The direction of the momentum of a body is the same as the direction in which it is moving.

The greater the momentum of a body, the greater its tendency to continue in motion. Thus, a baseball that is solidly struck by a bat (v large) is harder to stop than a baseball thrown by hand (v small), and an iron shot (m large) is harder to stop than a baseball (m small) of the same velocity.

Solved Problem 5.1 Find the momentum of a 50-kg boy running at 6 m/s.

Solution. The momentum can be calculated as follows:

mv = (50 kg)(6 m/s) = 300 kg·m/s

Impulse

A force F that acts on a body during time t provides the body with an impulse of Ft:

Impulse = Ft = (force)(time interval)

You Need to Know

The units of impulse are newton-seconds and pound-seconds.

When a force acts on a body to produce a change in its momentum, the momentum change m(v2v1) is equal to the impulse provided by the force. Thus

Ft = m(v2v1)

Impulse = momentum change

Solved Problem 5.2 A 46-g golf ball is struck by a club and flies off at 70 m/s. If the head of the club was in contact with the ball for 0.5 ms, what was the average force on the ball during the impact?

Solution. The ball started from rest, so v1 = 0 and its momentum change is:

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Conservation of Linear Momentum

According to the law of conservation of linear momentum, when the vector sum of the external forces that act on a system of bodies equals zero, the total linear momentum of the system remains constant no matter what momentum changes occur within the system.

Although interactions within the system may change the distribution of the total momentum among the various bodies in the system, the total momentum does not change. Such interactions can give rise to two general classes of events: explosions, in which an original single body flies apart into separate bodies, and collisions, in which two or more bodies collide and either stick together or move apart, in each case with a redistribution of the original total momentum.

Solved Problem 5.3 A rocket explodes in midair. How does this affect (a) its total momentum and (b) its total kinetic energy?

Solution.

(a) The total momentum remains the same because no external forces acted on the rocket.

(b) The total kinetic energy increases because the rocket fragments received additional KE from the explosion.

Collisions

Momentum is also conserved in collisions. If a moving billiard ball strikes a stationary one, the two move off in such a way that the vector sum of their momenta is the same as the initial momentum of the first ball (Figure 5-1). This is true even if the balls move off in different directions.

A perfectly elastic collision is one in which the bodies involved move apart in such a way that kinetic energy as well as momentum is con- served. In a perfectly inelastic collision, the bodies stick together and the kinetic energy loss is the maximum possible consistent with momentum conservation. Most collisions are intermediate between these two extremes.

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Figure 5-1

Solved Problem 5.4 A 2000-Ib car moving at 50 mi/h collides head-on with a 3000-lb car moving at 20 mi/h, and the two cars stick together. Which way does the wreckage move?

Solution. The 2000-lb car had the greater initial momentum, so the wreckage moves in the same direction it had.