Section 10–4 Momentum and energy

(Perfectly elastic collisions / Nearly elastic collisions / Rocket Propulsions)

In this section, the three interesting points are perfectly elastic collisions, nearly elastic collisions, and rocket propulsions. As an alternative, this section could be slightly revised and titled as “three types of collisions” (elastic, inelastic, and super-elastic). Historically, Wallis investigated perfectly inelastic collisions, Wren and Huygens focused on perfectly elastic objects, whereas Newton included experiments that are in between perfectly elastic and perfectly inelastic (French, 1971).

1. Perfectly elastic collisions:

“…the speeds before and after an elastic collision are equal is not a matter of conservation of momentum, but a matter of conservation of kinetic energy (Feynman et al., 1963, section 10–4 Momentum and energy).”

Feynman explains that the speeds before and after an elastic collision are equal is a matter of conservation of kinetic energy. It is worth mentioning that the total momentum of two objects is conserved whether the collision is perfectly elastic or perfectly inelastic. Importantly, there is a loss of kinetic energy for a brief moment during the impact when the two objects are in contact and both are compressed. Furthermore, both objects have zero velocity (in an inertial frame) when their kinetic energies are converted into potential energies of the elastic bodies. One may add that the internal forces (electromagnetic) cause the objects to decelerate and accelerate during the impact.

Feynman simply mentions that there are various degrees of elasticity, but he did not provide further details. Mathematically, it refers to the coefficient of restitution (e) between colliding objects and it is related to the ratio of the relative velocities of the two colliding objects after and before the collision: e = (v₂ – v₁)/(u₁ – u₂) in which the subscripts 1 and 2 refer to the object 1 and object 2 respectively. Physicists may define a perfectly elastic collision (or in short, elastic collision) as one where there is no loss of total kinetic energy after the collision. However, it is possible that the kinetic energies of both objects are converted into elastic potential energy momentarily during the collision.

2. Nearly elastic collisions:

“…between very elementary objects, the collisions are always elastic or very nearly elastic (Feynman et al., 1963, section 10–4 Momentum and energy).”

It is potentially confusing to students that there are so many terms such as “perfectly elastic collisions,” “perfectly inelastic collisions,” “elastic collisions,” “inelastic collisions,” and “nearly elastic collisions.” In general, physics teachers may explain that there is always a conversion of kinetic energy of a ball into thermal energy, elastic potential energy, and sound energy after a collision. On the other hand, Feynman states that the collisions between very elementary objects are always elastic (as an approximation) or very nearly elastic. In the real world, the collisions between atoms or molecules in a gas are said to be not perfectly elastic. For instance, there is a loss of kinetic energy in a collision of gas molecules due to an emission of infrared ray (or light).

Feynman suggests that it is feasible to make colliding bodies from highly elastic materials, such as steel (with carefully designed spring bumpers) such that a collision generates very little heat and vibration. Additionally, nearly elastic collisions are possible for systems that have no internal “gears, wheels, or parts” in which kinetic energy can be transferred. In 1964 (two years after this lecture of Feynman), Norm Stingley invented a toy ball (or Superball) that is made from a type of synthetic rubber instead of steel. Interestingly, Stingley offered the toy ball to his employer, Bettis Rubber Company in California, but it was turned down because it did not seem to be a profitable product. The toy balls are nearly elastic to the extent that they can bounce to about 90% (or up to 92%) of the drop height.

3. Rocket propulsion:

“Rocket propulsion is essentially the same as the recoil of a gun: there is no need for any air to push against (Feynman et al., 1963, section 10–4 Momentum and energy).”

Feynman ends the section by discussing rocket propulsion. He simplifies the calculation by using the law of conservation of momentum to deduce the velocity (v) of a rocket of mass (M) as equal to mV/M in which m represents a small piece of ejected mass that is moving with a velocity V relative to the rocket. Due to the continuous ejection of material, a more accurate equation would be mdV/dt = -vdm/dt.

More important, Tymms (2015) explains that “[t]he rocket effectively works by super-elastic collisions… (p. 131).” Essentially, rocket propulsion is contributed by super-elastic collisions in which the rocket’s kinetic energy is being increased by “explosions” (or chemical energy).

Students may be confused by Feynman’s statement that “there is no need for any air to push against” in order to have rocket propulsion. For example, one may use Newton’s third law to explain that there is an action that pushes the rocket forward (due to air) and a reaction on air (due to the rocket). One need not emphasize that air molecules continuously push the rocket forward, but they mainly move backward. Note that the law of conservation of momentum is more fundamental than Newton’s third law in the sense that it also holds true in quantum mechanics. In Feynman’s tips on physics, Feynman (2006) explains photon propulsion rockets as follows: “the momentum per second thrown out is the force needed to hold the rocket in place, while the energy per second thrown out is the power of the engine generating the photons (p. 88).”

Questions for discussion:

1. How would you define a perfectly elastic collision?

2. How would you explain that collisions in the real world are nearly elastic instead of perfectly elastic?

3. Is the principle of rocket propulsion is related to collisions?

The moral of the lesson: between very elementary objects, the collisions are very nearly elastic because the energy in the form of light or heat radiation could come out of a gas.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

2. 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.

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

4. Tymms, V. (2015). Newtonian Mechanics for Undergraduates. London: World Scientific.

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.