Grappling with variable mass problems

Mechanics courses often include physics problems with rocket launches. Because rockets eject mass as they launch, this scenario is a variable mass problem.

Correctly solving problems with variable mass is a difficult task, especially for a physics student trying to learn the best strategies. This same issue arises in problems involving chains and ropes sliding off tables or going around pulleys.

To compare approaches, Mungan and Lipscombe solved the vertical launch of a grappling hook using Newton’s Laws and an energy analysis. As the hook rises, it drags the rope behind it, increasing the system’s mass the higher it goes.

“Using Newton’s laws means thinking about the correct way to generalize F=ma,” said author Carl Mungan. “An energy analysis looks at what’s happening to the kinetic and potential energy as it goes up in the air. You can also compare that to what happens if the rope isn’t there.”

One important aspect of this problem is defining the system. Some approaches include the entire rope, while others include just the component of the rope in the air. When considering the rope in the air, the length must increase, or the mass must change as the hook rises.

“We compare these methods and discuss how you would approach the problem in each of these ways,” said Mungan. “We wanted to find out what was the easiest way for a student to think about it and which ways lead to pitfalls.”

While these solutions can be applied to mountain climbing or scaling a wall, the team thinks it will be most useful in the classroom.

Source: “Vertical launch of a grappling hook,” by Carl E. Mungan and Trevor C. Lipscombe, The Physics Teacher (2022). The article can be accessed at https://doi.org/10.1119/5.0030313 .