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Conservation of Momentum

Conservation is one of those words in physics which has a special meaning. When a physicist says that momentum is conserved, she means that momentum ("movage") has a kind of permanence, that it cannot be created from nothing nor destroyed. The formal statement is in the box:
If a system of objects is isolated (external forces are balanced), then the total momentum of the system stays constant over time, that is, the total momentum is constant. In particular,



An external force is a force on one of the objects in the system by an outside agent, while an internal force is a force between two objects in the system.


A Mack truck (9000 kg) going north at 10 m/s encounters a Porsche (1000 kg) going south at 20 m/s. What is the velocity (speed and direction) of the resulting fused mass of metal?



We DRAW A DIAGRAM (Figure 8-3). Let's discuss the collision itself, shown in the middle part of Figure 8-3. There are two normal forces and two gravitational forces which are all external. These, however, are balanced. The force of the truck on the car and that of the car on the truck are internal forces.


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Figure 8-3

Momentum is conserved, and we write (in one dimension)

Pbefore = Pafter


mMvM + mpvp = (mM + mp)vf


(9000 kg)(10 m/s) + (1000 kg)(–20 m/s) = (10,000 kg)vf


vf = 7 m/s


Notice we used vp = –20 m/s, since it was going south, and we chose north to be positive. We have to pay attention to signs because momentum is a vector quantity.



During the collision in Section A, the external forces are the tensions in the threads and gravity, and these are balanced. The internal forces are all the complicated forces among the balls. So momentum is conserved. We do not yet know enough to show why exactly one ball jumps off the right end. Momentum would be conserved, for example, if all five balls headed to the right at one fifth the impact velocity of the left ball.



Is momentum conserved while the left ball is swinging from its initial height on its way to collision? (See Figure 8-4.)


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Figure 8-4


It certainly does not seem so, since the ball starts with zero momentum and achieves a maximum momentum just before impact. Figure 8-5 shows why momentum is not conserved. The external forces on the ball are unbalanced.


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Figure 8-5



Is momentum conserved for a crocodile dropped from a ladder? If not, what is the external force? Try doing this one yourself. (See Figure 8-6.)


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Figure 8-6

Here is a major hint:
Whenever a problem involves a collision, especially one with crunching, crashing, or sticking, you will probably need to use conservation of momentum.



A car (1000 kg) going north (10 m/s) collides with a truck (1500 kg) going east (20 m/s). Assume there is negligible friction at the time of the collision. What is the final speed and direction of the combined car/truck?



We DRAW A DIAGRAM (Figure 8-7).


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Figure 8-7

The total momentum before the collision can be read from Figure 8-8.

From the Pythagorean theorem we find the magnitude of the total momentum ptot = 3.2 x 104 kg m/s. The magnitude of the velocity is v = (3.2 x 104 kg m/s)/(2500 kg) = 13 m/s. The direction we obtain from tanφ = (1 x 104 kg m/s)/(3 x 104 kg m/s). Thus φ = 18˚ north of east.


Figure 8-8


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