Change in momentum is force on a good day

The day dawned clear, promising to be hot, and the sun delivered the heat. I was well prepared to attempt to duplicate the one off success of spring 2019. The demonstration that the force applied to a moving system is equivalent to the change in momentum is a demonstration that has a long history of failure.

My earliest attempts entangled energy considerations which obscured the role of momentum. Other very early attempts tried weights dropping from cross-members of the covered walkway - friction meant the weights did not fall back down and the result was hazardous. I also tried variations on pulling from early on, but often met with failure. A failure fall 2018 was due to a student holding the piston of the spring scale, this actually occurred again this term. When told the spring scale should read 20 Newtons, the students oblige by setting the scale to 20 Newtons and then holding both ends of the scale. I also failed to make clear that the student had to run backwards at an accelerating rate to maintain the 20 Newton pull. This lead to numerous restarts as I tried to clarify how to maintain a constant pulling force.

Prior to run I remembered to mass both myself and the board at the same time. The board add two to three kilograms to the overall mass.

I began in the classroom. While I mentioned that failure was an option, I was optimistic enough not to put that on the board in writing this term. Due to Liberation day falling on "kinetic energy Wednesday" of week four, I had dropped the kinetic energy demonstration this term. Week four focused on momentum only. So I could not launch force week from an energy perspective, nor would that have helped. Conservation of momentum had arisen from laboratory four, and that provided a jumping off point for this week.

If there is nothing pulling or pushing on a system, then momentum is conserved. Essentially Newton's first law - an object retains its momentum. Those at rest remain at rest. Those in motion remain in motion. Since Newton's laws do not enter the curriculum until Wednesday, I did not make this connection today, but perhaps I should have. If something is pulling or pushing on a movable object, then the momentum will change. And that change in momentum will be equal to the pushing or pulling, to the force on the system. The change in momentum with respect to time will be equal to the force.

This term I did not write the d(p)/dt = d(mv)/dt and remained with delta notation only. Force = Δ(mv)/Δt

Then I laid out the proposed demonstration of this idea. The demonstration takes some gear, primarily multiple timers to handle the times that need to be recorded along with spring scales and a tape measure. A tow rope and RipStik are also necessary.

This term I used six timers, three on each segment with the intent of using median times. I used the white spring scale only as 20 N is near the middle of that scale.

Start of the run was from the usual zero mark on the sidewalk

My mass plus the mass of the RipStik was 69.3 kg. I started the RipStik at a speed of zero meters per second and was accelerated across a distance of three meters by a rope being pulled at a fairly constant 20 Newtons. Both fall 2018 and spring 2019 I had used 20 Newtons, and that had seemed a reasonable pulling force, one that I could absorb and transfer to the RipStik deck without being yanked off at the get go.

End of the speed trap

This term a two meter speed trap was again used after I dropped the rope at the three meter mark to determine my final velocity. Fall 2018 a one meter trap was tad too short. A three meter trap would give the timers more time, but perhaps I would slow down too much during that time.

The set up on the sidewalk. Some of the chalk lines are from the morning statistics linear regression exercise. I hold the rope for the first three meters with 20 Newtons tension on the rope, then I coast for 2 meters in the speed trap.

Photos were captured by a student, here one can see that the rope pulling student is holding both the outer barrel of the spring scale and the plunger, fixing the scale at 20 Newtons and rendering the pull tension unknown.

In this second image this issue is sorted out and the student is holding only the barrel.

Holding the rope in my left hand, I held on to the zero post. At first I called the start using a 3-2-1, go! sequence. The problem was that the puller lost 20 Newtons before they could react and start back peddling. So I told the puller to call the start, which seemed to actually work better because all I have to do is let go of the post and try to stay up on the board. At very low speed. I would recommend future attempts have the puller call the start.

Here I am in the first three meters under acceleration from the tow rope.

Three timers timed the three meter accelerating run, and another three timed the two meter "speed trap." The data can be seen in the diagram above. While the 20 Newton force is not strictly equal to the various momentum values derived from the various timers, the difference is as small as 18% based on one of the timings. The class was running out of time in the class period, I suspect that a couple more successful runs might have helped. At this point the puller had swapped out for another student who seemed more confident they understood what was necessary on the pulling end of the operation.

Thus while not strictly equal, the values are roughly and reasonably equivalent given the manner in which the data was generated. As one student noted, "The scale bounces back and forth." Thus the force was perhaps more than 20 Newtons at some points in the run.

The constraints of time meant that I did not have the students running the calculations as I did last spring. Perhaps a variation of the above diagram with blanks for the time and momentum calculations, having the students work from the different timers, would help put the students back into the calculation loop given the brevity of the period.

While the result was not as stunningly equal as spring 2019 the demonstration was not the orders of magnitude errors that have been seen in the past. The basic structure is functional. I still feel that back peddling to maintain 30 Newtons would be too challenging for the puller, although the acceleration would be brisker. I suspect I could handle 30 Newtons, both in transferring that force to the RipStik deck and in terms of the final velocity. Perhaps a bit more explanation of us, the puller and I, moving as an accelerating system would help in any future iterations. If nothing else, this is one of the most fun demonstrations for me to engage in.

Photo credits go to Gregorlyn.