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9-4 Conservation of Linear Momentum

In the previous section we discussed the fact that an impulse applied to an object causes a change in the object's momentum. Consequently, if there is no impulse then there is no change in momentum. The net impulse on an object will be zero when the net force on that object is zero. This leads to the following statement:

When the net force on an object is zero, its linear momentum is conserved.

The situation becomes a little more interesting for a system of objects such as the billions of air molecules in an enclosed container such as your bedroom. In this latter case you have billions of particles interacting with each other that cancel out when the system is considered as a whole. That is, the sum of all forces internal to the system is zero, .

We are left, therefore, with only the external forces applied to a system to make up the overall net force . By applying Newton's second law, we can see that the net impulse on a system equals the change in its total momentum p_total (or p_net)

.

By similar reasoning as in the single particle case, we can see from this expression that

if the net external force on a system is zero then the total linear momentum of the system is conserved.

When the total momentum of a system is conserved we can often apply this by setting the final total momentum equal to the initial total momentum

Notice that because momentum is a vector quantity it, unlike energy (a scalar), must be conserved in both magnitude and direction.

Practice Quiz

Which of the following statements is most accurate? The linear momentum of a system is always conserved. The linear momentum of a system is only conserved if no external forces act on the system. The linear momentum of a system is conserved if the net external force on the system is zero. The linear momentum of a system is not conserved. [none of the above]

9-5 Inelastic Collisions

When studying collisions, we often divide them into two categories according to whether or not the total kinetic energy of the colliding bodies is conserved. If the total kinetic energy is not conserved, the collision is called an inelastic collision. In analyzing these types of collisions we usually just apply the conservation of the total linear momentum to the system p_f = p_i. A special case of an inelastic collision occurs when the colliding objects stick together and emerge from the collision effectively as one object. This latter case is called a completely inelastic collision because in this case the system loses the maximum amount of kinetic energy that is can lose while still conserving momentum.

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Physlet Illustration: Collision in One Dimensions
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A 100-gm red ball and a 75-gm blue ball collide. Is momentum conserved?  Is the collision elastic or inelastic? Start
Hints: Determine each ball's velocity before and after the collision. (Remember velocity is a vector.) Using these velocities, as well as the mass of each ball, calculate each ball's momentum before and after the collision. Using the masses and velocities before and after, calculate the kinetic energy before and after.

Example 9.4 One-dimensional collision: The driver of a 1485 kg car is driving along a street at 25 mi/h. Another driver, coming from directly behind (and not paying attention), is driving a 1520 kg car at 38 mi/h and hits the slower car. If both drivers slam the brakes at the moment of impact and their fenders catch on to each other, with what combined speed do they begin to skid?

Picture the Problem The upper picture shows the heavier and faster car (m₁) catching up to the lighter car (m₂). The lower picture shows them stuck together after the collision.

Strategy This is a completely inelastic collision. We apply the conservation of momentum by setting the total momenta before and after the collision equal. Take the direction of motion to be the x-direction.

Solution

1. The total momentum before the collision:	ptotal = m1v1 + m2v2
2. The total momentum after the collision:	ptotal = (m1 + m2)v
3. Setting them equal gives:
4. The numerical result is:

Insight Since this was just a one-dimensional problem we dispensed with full vector notation.

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Practice Quiz

A cart is rolling along a horizontal floor when a box is dropped vertically into it. After catching the box the cart will move... more slowly. more quickly. at the same speed as before. in a different direction. [the cart will come to a stop as a result of catching the box.]

Two objects of equal mass of 1.7 kg move directly toward each other with equal speed of 2.2 m/s. If they stick together after the collision their speed will be... 0.65 m/s 2.2 m/s 1.1 m/s 3.7 m/s 0 m/s

9-6 Elastic Collisions

A second category of collisions considers those for which the total kinetic energy is conserved. These collisions are called elastic collisions. Keep in mind that, as with momentum, it is the total kinetic energy of the system, not of any individual particle, that we are considering

In everyday life few collisions are precisely elastic. However, many everyday collisions are approximately elastic because such a small fraction of the kinetic energy is lost. Microscopically, perfectly elastic collisions are commonplace. Along with the conservation of kinetic energy, we still must apply the conservation of momentum to these collisions as well.

In one dimension (a head-on collision), we can combine the equations for the conservation of momentum and the conservation of kinetic energy and solve them for the final velocities of the two particles. For particles m₁ and m₂ with m₂ initially at rest and m₁ moving with speed v₀ directly toward m₂, the result is

.

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Example 9.5 Head-on elastic collision: A 1.3 kg ball initially moving with a speed of 3.1 m/s strikes a stationary 2.2 kg ball head-on. What are the final velocities of the two balls?

Solution: This is precisely the situation discussed above. Let's identify the masses and velocities.

Given: m₁ = 1.3 kg, v₀ = 3.1 m/s, m₂ = 2.2 kg Find: v_1f, v_2f

Making direct use of the equations listed above we have

Insight Notice that after the collision the balls move in opposite direction.

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Physlet Illustration: Collision in Two Dimensions
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A 100-gm red ball and a 50-gm blue ball collide. Can you verify that momentum is conserved in the collision? Is the collision elastic or inelastic? Start
Hints: Determine the x and y components of each ball's velocity before and after the collision. Using these components of velocity, as well as the mass of each ball, calculate the x and y components of each ball's momentum before and after the collision. Using these velocities and masses, calculate the kinetic energy before and after the collision.

Example 9.6 Elastic collision in two-dimensions: Two objects, one (m₂) of mass 1.50 kg moving due south at 15.3 m/s and the other (m₁) of mass 1.19 kg moving northeast at 11.7 m/s, collide elastically. After they collide, m₂ moves with a speed of 12.3 m/s at 10.9^o north of east. What is the final velocity of m₁?

Picture the Problem The picture shows the two masses moving toward each other before colliding as well as the speed and direction of m₂ afterward.

Strategy To perform the analysis we use the conservation of kinetic energy together with the conservation of momentum in both the x- and y-directions.

Solution

1. Apply the conservation of kinetic energy:
2. Insert numerical values and solve for v1f:
3. Apply the conservation of momentum along the x-direction:
4. Evaluate p1f,x:
5. Apply the conservation of momentum along the y-direction:
6. Determine the reference angle:
7. Since both components are negative q1f,ref is measured from the negative x-axis in the 4th quadrant. Therefore, the final velocity of m1 is:	15.5 m/s at 63.5o south of west.

Insights This problem had many steps but we started with the conservation of kinetic energy to insure that we are describing an elastic collision. Also, it was helpful to use the equations for the conservation of momentum in both the x- and y-directions to make sure that we had the correct orientation of v_1f.

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Practice Quiz

When a lighter mass undergoes a head-on, elastic collision with a heavier mass that is initially at rest, the lighter mass will... stop and come to rest continue forward at a slower speed recoil and move in the opposite direction to its original motion. [cannot answer definitively, it depends on the masses and the initial speed of the lighter mass] [None of the above.]

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