Day 07 - Drag Forces

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Announcements

• Homework 2 is due Friday (late Sunday)

• Homework 3 is now posted

• Office hours, for now (Mihir-MN; Danny-DC):

  • Thursday 3-5pm (MN, 1248 BPS)

  • Friday 10-12pm (DC, 1248 BPS)

  • Friday 3-5pm (MN, Zoom only)

• Zoom Link for class: https://msu.zoom.us/j/99550311023

  • password: phy321msu

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Seminars this week

WEDNESDAY, January 28, 2026

Astronomy Seminar, 1:30pm, BPS 1400 & Zoom Speaker: Rebecca Kyer & Emily Elizondo, MSU Title: TBA Zoom Link: https://msu.zoom.us/j/93334479606?pwd=OtIXPWhRPBfzYu53sl3trSJlaBYI7C.1 Passcode: 825824

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Seminars this week

WEDNESDAY, January 28, 2026

Physics Education Research Seminar, 3:00pm, BPS 1400 & Zoom Speaker: Jennifer Doherty, MSU Title: Principle-based reasoning: A strategy for developing expertise in Physiology Zoom Link: https://msu.zoom.us/j/96470703707 Passcode: PERSeminar

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Seminars this week

WEDNESDAY, January 28, 2026

FRIB Nuclear Science Seminar, 3:30pm, Zoom Only Speaker: Gwen Grinyer, University of Regina Title: Precision Spectroscopy of Rare Isotopes Zoom Link: https://msu.zoom.us/j/91051885898?pwd=S423bks5tzaeOwNb1tLaUaHDUScm5A.1 Passcode: 949110

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Seminars this week

FRIDAY, January 30, 2026

QuIC, Seminar, 12:40 p.m., BPS 1300 & Zoom Speaker: Ben DalFavero, MSU Title: Fault tolerant quantum computing II *For the full schedule, please see: https://sites.google.com/msu.edu/quic-seminar/ or for more information, please reach out to Ryan LaRosa directly

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Seminars this week

FRIDAY, January 30, 2026

IReNA Online Seminar, 2:00 pm, Zoom Light refreshments at 1:50pm in 2025 Nuclear Conference Room - FRIB Hosted by: Aldana Grichener (University of Arizona & Observatory) Speaker: Mengke Li, University of California, Berkeley Title: Implications of a Weakening N = 126 Shell Closure Away from Stability for r-Process Astrophysical Conditions Zoom Link: https://msu.zoom.us/j/827950260 Password: CENAM

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Goals for Week 3

• Be able to answer the following questions.

  • What is Mathematical Modeling?

  • What is the process for analyzing these models?

• Be able to solve “Simple” Motion Problems with Newton’s Laws.

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Our man, Reynolds

The Reynolds number is a dimensionless quantity. It is a ratio of inertial forces to viscous forces.

\[Re = \frac{\rho v L}{\mu}\]

• \(\rho\) - density of the fluid

• \(v\) - velocity of the object

• \(L\) - characteristic length

• \(\mu\) is the dynamic viscosity

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Our man, Reynolds

BTW, this is not a photo of Reynolds. This is Stokes.

• He developed the concept of the Reynolds number.

• Reynolds “popularized” it according to the Wikipedia.

\[Re = \frac{\rho v L}{\mu}\]

Discussion: What kinds of systems have a high/low Reynolds number?

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Clicker Question 7-1

Consider a ball falling down with a coordinate system such that \(y\) is positive downward. Which of the following is correct for the differential equation?

1. \(m\ddot{y}=-mg+bv_y\)

2. \(m\ddot{y}=-mg-bv_y\)

3. \(m\ddot{y}=+mg+bv_y\)

4. \(m\ddot{y}=+mg-bv_y\)

Think about the SIGN of \(v_y\); it’s a component not a speed!

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Assuming a linear model for Air Resistance \(\sim bv\), we can obtain this EOM for the velocity a falling ball, after separation of variables:

\[\dfrac{dv_y}{v_y - v_{term}} = -\frac{b}{m}dt\]

Let’s try to integrate this equation from \(v_y = v_{y0}\) at \(t=0\) to \(v_y(t)\) at time \(t\).

We can write the integral (on both sides) that is needed to solve this equation.

This integral setup is common in physics derivations with ODEs; provided we can find an anti-derivative.

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Clicker Question 7-2

Assuming a quadratic model for Air Resistance \(\sim cv^2\), what is the EOM for a falling ball, with \(y\) positive upward?

1. \(m\ddot{y}=-mg+c v_y^2\)

2. \(m\ddot{y}=-mg - c v_y^2\)

3. \(m\ddot{y}=+mg + c v_y^2\)

4. \(m\ddot{y}=+mg - c v_y^2\)

5. ???

Draw a picture, remember \(v_y\) is a component, and consider upward and downward motion!

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Clicker Question 7-3

For the system of Quadratic Drag in 1D, we found a solution for the velocity as a function of time, with \(v = 0\) at \(t = 0\).

\[v(t) = v_{term}\tanh(gt/v_{term})\]

where \(v_{term} = (mg/c)^{1/2}\). Do the units make sense? What are the units \(\left[gt/v_{term}\right]\)?

1. Yes, they have the same units; the units for \(\left[gt/v_{term}\right]\) are m/s.

2. No, they have different units; the units for \(\left[gt/v_{term}\right]\) are m/s.

3. Yes, they have the same units; the units for \(\left[gt/v_{term}\right]\) are unit-less.

4. No, they have different units; the units for \(\left[gt/v_{term}\right]\) are unit-less.

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Clicker Question 7-4

For the system of Quadratic Drag in 1D, we found a solution for the velocity as a function of time, with \(v = 0\) at \(t = 0\).

\[v(t) = v_{term}\tanh(gt/v_{term})\]

where \(v_{term} = \sqrt{mg/c}\). What happens when \(t \rightarrow \infty\)?

1. The object stops moving.

2. The object travels at a constant velocity.

3. The object travels at an increasing velocity.

4. The object travels at a decreasing velocity.

5. I’m not sure.

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Clicker Question 7-5

Assuming a quadratic model for Air Resistance \(\sim cv^2\), what is the EOM for a ball initially sent upward and to the right, with \(y\) positive upward and \(x\) positive to the right?

1. \(m\ddot{x}=-c v_x\sqrt{v_x^2+v_y^2}; \quad m\ddot{y}=-mg - c v_y\sqrt{v_x^2+v_y^2}\)

2. \(m\ddot{x}=-c v_x\sqrt{v_x^2+v_y^2}; \quad m\ddot{y}=+mg - c v_y\sqrt{v_x^2+v_y^2}\)

3. \(m\ddot{x}=+c v_x\sqrt{v_x^2+v_y^2}; \quad m\ddot{y}=-mg + c v_y\sqrt{v_x^2+v_y^2}\)

4. \(m\ddot{x}=+c v_x\sqrt{v_x^2+v_y^2}; \quad m\ddot{y}=+mg + c v_y\sqrt{v_x^2+v_y^2}\)

5. None of the above.

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Clicker Question 7-6

For linear drag in 2D, we found the following equations of motion:

\[\dot{v}_x = -\gamma v_x\]

\[\dot{v}_y = -g - \gamma v_y\]

Are these equations integrable? Can we find anti-derivatives for both equations?

1. Yes, both equations are integrable.

2. No, neither equation is integrable.

3. Only the \(x\) equation is integrable.

4. Only the \(y\) equation is integrable.