Section 10–3 Momentum is conserved

(Center of mass frame / Laboratory frame / Moving car’s frame)

In this section, Feynman discusses collisions by comparing “view from moving car” with “view from center of mass,” “view from laboratory,” and “view from CM system”. Essentially, the discussions include three reference frames: center of mass frame, laboratory frame, and moving car’s frame. The title of this section could be rephrased as “three reference frames of collisions”.

1. Center of mass frame:

“Two views of an inelastic collision between equal masses (Feynman et al., 1963, section 10–3 Momentum is conserved).”

Feynman mentions the need for an assumption that physical laws remain the same whether we are standing still or observing in a moving car. This assumption is applicable to collisions such as two moving objects collide and stick together. The collision in the example is also physically equivalent to an object, initially moving with velocity 2v, hits a stationary object that has the same mass. Interestingly, Feynman shows how the velocities of the two objects (after a collision) can be deduced without using the concept of linear momentum. In essence, he uses simple logic and two different frames of reference: center of mass frame and a moving car’s frame.

In Fig. 10.4, the phrase “view from center of mass” is used to describe a perspective from the center of mass frame. This frame of reference refers to an inertial frame in which the center of mass is fixed at the origin. Center of mass frame is also known as “zero momentum frame” in the sense that the total linear momentum of a system of particles is zero (Williams, 2002). An advantage of zero momentum frame is the convenience of viewing the collision such that it simplifies the calculations. Mathematically, if the first particle approaches the collision point (or the second particle) with linear momentum p, then the second particle will approach the same point with linear momentum -p.

2. Laboratory frame:

“When it is all finished they will be moving at ½(v₁−v₂) with respect to the car. What then is the actual speed on the ground? (Feynman et al., 1963, section 10–3 Momentum is conserved).”

In another inelastic collision, we assume that two identical objects initially move with velocities v₁ and v₂ before they stick together. If we observe the collision in a car, say moving at velocity v₂, one of the objects would appear to be at rest and the other would appear to have a velocity of v₁−v₂. After the collision, they are expected to be moving at an average velocity of ½(v₁−v₂) with respect to the moving car or ½(v₁−v₂) + v₂ with respect to the ground. Feynman uses Galilean principle of relativity again to deduce the final velocity of the two objects without first using a mathematical formula involving linear momentum. Note that he uses the laboratory frame and moving inertial frame instead of the center-of-mass frame in this example.

In Fig. 10.5, the phrase “view from lab” is used to describe the laboratory frame of reference. In short, we may use the term laboratory frame or Earth’s frame to describe a frame of reference that views from a laboratory on the Earth in which an experiment is performed. For reasons of simplicity, physicists may assume the laboratory frame to be an inertial frame that is at rest (French, 1971; Morin, 2008). In particle physics, one may consider the laboratory frame in which the detectors for a particle accelerator are stationary with respect to the Earth. To be more precise, physicists may state that the laboratory is rotating with the Earth about its axis. Thus, a laboratory frame can also be defined as a non-inertial frame or a rotating frame of reference (Ryder, 2008).

3. Moving car’s frame:

“This is very easy to answer using our principle of Galilean relativity, for we simply watch the collision which we have just described from a car moving with velocity −v/2 (Feynman et al., 1963, section 10–3 Momentum is conserved).”

Feynman discusses another inelastic collision problem in which the two objects have different masses. The question is what will happen if an object that has mass m moves with velocity v hits a stationary object that has mass 2m? Based on Feynman’s thought experiment, the masses m and 2m attain velocities –v and v/2 respectively. He uses “view from CM system” (instead of “view from center of mass”) first followed by “view from car.” In other words, it is insightful to visualize and solve the collision problem by using the “zero momentum frame” and “moving car’s frame.” Perhaps Feynman could be more explicit and explain why he adopts the two different reference frames.

In Fig. 10.7, the phrase “view from car” is used to describe an inertial frame of reference based on an observer that is in a car moving at constant velocity. We may use the term “moving car’s frame” to describe an inertial frame in which an observer views one of the objects to be stationary (as a target). Thus, the momentum of an object becomes zero and it slightly simplifies the calculation needed. Interestingly, a definition of an inertial frame is based on the validity of Newton’s first law (Kleppner & Kolenkow, 2014). Philosophers may question this definition because Newton’s laws are also defined in terms of the inertial frame. More importantly, Feynman has explained that the conservation of momentum is supported by an experiment using an air trough (instead of dependent on how we define the inertial frame).

Questions for discussion:

1. What is/are the advantage(s) of using a center of mass frame of reference (or zero momentum frame)?

2. What is/are the advantage(s) of using a laboratory frame of reference (or Earth’s frame)?

3. What is/are the advantage(s) of using a moving car’s frame of reference?

The moral of the lesson: it is possible to deduce the speed of an object after a collision by using three different frames of reference and without involving calculations based on the law of conservation of momentum.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Kleppner, D., & Kolenkow, R. (2014). An Introduction to Mechanics (2nd ed.). Cambridge: Cambridge University Press.

4. Morin, D. (2008). Introduction to Classical Mechanics. Cambridge: Cambridge University Press.

5. Ryder, P. (2007). Classical Mechanics. Aachen: Shaker Verlag.

6. Williams, W. S. C. (2002). Introducing Special Relativity. London: Taylor & Francis.