Day 04 - Mathematical Preliminaries

Day 04 - Mathematical Preliminaries#

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Announcements#

  • Homework 1 is due this Friday

  • Homework 2 is posted now

  • Help sessions start this week

    • DC Friday at 2-4pm (1248 BPS)

  • Mihir (ULA) will host additional help hours soon

Seminars this week (Wednesday)#

WEDNESDAY, September 3, 2025#

Astronomy Seminar, 1:30 pm, 1400 BPS, In Person and Zoom, Host~ Speaker: Group Introductions Title: Zoom Link: https://msu.zoom.us/j/887295421?pwd=N1NFb0tVU29JL2FFSkk0cStpanR3UT09 Meeting ID: 887-295-421 Passcode: 002454

Seminars this week (Wednesday)#

WEDNESDAY, September 3, 2025#

Nuclear Science Seminar, 3:30Pm., FRIB 1300 lab in person and online via Zoom Speaker: Mark Spieker, Florida State University Title: Experimental studies of the pygmy dipole resonance Zoom Link: https://msu.zoom.us/j/95277003505?pwd=hTILu1oLqmhTCU7jlVKFTlXXZBmuGb.1 Meeting ID: 952 7700 3505 Passcode: 404830

Goals for this week#

Be able to answer the following questions.#

  • What are the essential physics models for single particles?

  • How do we setup problems in classical mechanics?

  • What mathematics do we need to get started?

  • How do we solve the equations of motion?

Reminders from Day 03#

  • In a Newtonian world, we start from a vector description of motion

  • Differential equations are mathematical models that describe the motion of particles

  • We can use different methods to solve these differential equations

i-Clicker: https://join.iclicker.com/QTEC

Projectile Motion#

\[\mathbf{a} = \langle a_x, a_y \rangle\]
\[x_f = x_i + v_{x,i}t + \dfrac{1}{2}a_x t^2\]
\[y_f = y_i + v_{y,i}t + \dfrac{1}{2}a_y t^2\]

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Clicker Question 4-1#

For this fountain, what is the best guess for the acceleration (\(\mathbf{a} = ??\)) experienced by a fluid particle?
Assume \(y\) is positive upward; \(x\) is positive to the right.

  1. \(a_x \neq 0, a_y = g\)

  2. \(a_x = 0, a_y = g\)

  3. \(a_x \neq 0, a_y = -g\)

  4. \(a_x = 0, a_y = -g\)

  5. Something else

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Clicker Question 4-2#

The average velocity for a macroscopic time step \(\Delta t = t_f - t_i\) is given by:

\[\mathbf{v}_{avg} = \dfrac{\Delta \mathbf{r}}{dt}\]

where \(\Delta \mathbf{r} = \mathbf{r}_f - \mathbf{r}_i\). At what time do we estimate the average velocity occurs?

  1. \(t_i\)

  2. \(t_f\)

  3. Sometime between \(t_f\) and \(t_i\)

  4. \(\dfrac{t_f-t_i}{2}\)

Clicker Question 4-3#

Consider the generic position vector \(\vec{R}\) for a particle in 2D space. Which of the following describes the direction of the vector in plane polar coordinates (\(r\), \(\phi\))?

  1. \(\hat{R}\)

  2. \(\hat{r}\)

  3. \(\hat{\phi}\)

  4. Some combination of \(\hat{r}\) and \(\hat{\phi}\)

  5. I’m not sure.

Group Discussion 4-1#

We found the following expression for the equation of motion of a falling ball subject to air resistance:

\[m \ddot{y} = +mg - b \dot{y} - c \dot{y}^2\]

What are the units of the constants \(b\) and \(c\)?

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