Two Chapter 1

Conservation of Momentum

• Newton's third law tells us that:

• In words, the force of object "a" on object "b" is equal in magnitude but opposite in direction to the force of object "b" on object "a".
• Continuing this reasoning we have:

• This conservation of momentum can be experimentally proven.
  • Conservation of momentum is not limited to two objects. Any number of objects interacting will display conservation of momentum.
• The Law of Conservation of Momentum - If the net external force acting on any object or system of objects is zero, then the total momentum of the object or system of objects remains constant.

• Isolated System - A system in which no net external force acts on the system.
  • EG. a ball falling to the earth is an isolated system
    • A ball by itself is not isolated, since the Earth pulls on it.
• Conservation of Momentum has many applications
  • Gas molecules collide with the container walls to produce pressure
  • Mass of subatomic particles can be determined through collisions.
  • Atomic structure has been determined through collisions.
  • The Moon and Earth interaction causes tides.
  • Rockets operate by the principle of Conservation of momentum.
    • Rocket problems are complicated because
      • the mass of the rocket is not constant (decreases as gas is released)
      • The velocity of the gas ejected with respect to the rocket remains constant, and the velocity of the rocket can exceed the relative velocity of the exhaust gas.
• Thrust - forward force on an object such as a rocket or air plane.

• F_t = forward thrust, V_g = exhaust speed with respect to the rocket, and Dm_g/DT = the rate at which mass is ejected.

The Vector Nature of Momentum

• So far we have looked at collisions along a line.
• Consider a collision between two air pucks as on an air hockey table.
  • See the data by clicking Here.
• The time between dots in the picture is 0.030 s
• The mass of the air pucks are 0.548 kg and 0.551 kg as shown
• The 0.551 kg puck was stationary to start with.
• The incident puck comes from the bottom and glances off to the right.
• On my screen, the picture is to full scale, but just in case it is not to full scale on your screen, the diameter of the pucks is 7.75cm. The distance A is 3.90 cm, distance B is 3.24 cm and the distance C is 3.34 cm. From this information we can do some calculations.
  • We first determine the velocities before and after collision.

• The velocities of the two objects after collision are now calculated.

• Now we determine the momentum of each object before and after interaction.

• Momentum is a vector quantity, and to see if momentum is conserved, we must break each of these up into their x and y components.
• We will choose the original direction of object 1 (0.548 kg object) to be the + y direction.
• The angle the +y direction makes with the direction of object 1's motion after interaction is 41.3^o.
• The angle the + y direction makes with the direction of object 2's (0.551 kg) motion after interaction is 27.4^o.
• We now calculate the x and y components of momentum for each object before collision.

• Now we calculate the momentum components of each object after collision.

• When we consider the limits of experimental error, we can conclude that momentum is conserved in two dimensional collisions.
• Another way to state the above is:
  • The vector sum of all momentums before interaction equals the vector sum of all momentums after interaction.
  • In mathematical form:

• Working with the first of these:

• This last equation tells us that for two objects interacting, the change in momentum of one object, is equal to the change in momentum of the other, only in the opposite direction.
• Rearranging the last equation above gives:

DP₁ + DP₂ = 0 = DP_total

• From this we conclude that the total change in momentum in an isolated system is zero
• It does not matter what the circumstances are, how many objects are present, whether one dimensional or multidimensional, what the masses are momentum in all dimensions is conserved!
• The law of conservation of momentum applies to any isolated interactions
• Kinetic energy does not have to be conserved in these interactions, though it may be conserved.

November 23, 2013