Events on Spacetime Diagrams

  • Special Relativity Spacetime Diagram Simultaneity Colocation
    • group Events on Spacetime Diagrams

      group Small Group Activity

      5 min.

      Events on Spacetime Diagrams
      Theoretical Mechanics 2021

      Special Relativity Spacetime Diagrams Simultaneity Colocation

      Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.
    • accessibility_new Time Dilation Light Clock Skit

      accessibility_new Kinesthetic

      5 min.

      Time Dilation Light Clock Skit

      Special Relativity Time Dilation Light Clock Kinesthetic Activity

      Students act out the classic light clock scenario for deriving time dilation.
    • assignment_ind Time Dilation

      assignment_ind Small White Board Question

      10 min.

      Time Dilation
      Theoretical Mechanics (4 years)

      Time Dilation Proper Time Special Relativity

      Students answer conceptual questions about time dilation and proper time.
    • face Lorentz Transformation (Geometric)

      face Lecture

      30 min.

      Lorentz Transformation (Geometric)
      Theoretical Mechanics (3 years)

      Special Relativity Lorentz Transformation Hyperbola Trig

      In this lecture, students see a geometric derivation of the Lorentz Transformation on a spacetime diagram.
    • assignment Gibbs entropy is extensive

      assignment Homework

      Gibbs entropy is extensive
      Gibbs entropy Probability Thermal and Statistical Physics 2020

      Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

      1. Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
      2. Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

      In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

    • group Right Angles on Spacetime Diagrams

      group Small Group Activity

      30 min.

      Right Angles on Spacetime Diagrams
      Theoretical Mechanics (4 years)

      Special Relativity

      Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.
    • assignment Scattering

      assignment Homework

      Central Forces 2023 (3 years)

      Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

      1. What is the initial angular momentum of the system?
      2. What is the initial total energy of the system?
      3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
      4. Sketch the effective potential.
      5. What is the angular momentum at \(r_{\rm{min}}\)?
      6. What is the total energy of the system at \(r_{\rm{min}}\)?
      7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
      8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
      9. What is the value of the effective potential at \(r_{\rm{min}}\)?
      10. For what values of the initial total energy are there bound orbits?
      11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

    • group Mass is not Conserved

      group Small Group Activity

      30 min.

      Mass is not Conserved
      Theoretical Mechanics (4 years)

      energy conservation mass conservation collision

      Groups are asked to analyze the following standard problem:

      Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

    • assignment Effective Potentials: Graphical Version

      assignment Homework

      Effective Potentials: Graphical Version
      Central Forces 2023 (2 years)

      Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).

      1. Give the definition of a central force system and briefly explain why this situation qualifies.
      2. Make a sketch of the graph of the effective potential for this situation.
      3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
      4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

    • grading Free expansion

      grading Quiz

      60 min.

      Free expansion
      Energy and Entropy 2021 (2 years)

      adiabatic expansion entropy temperature ideal gas

      Students will determine the change in entropy (positive, negative, or none) for both the system and surroundings in three different cases. This is followed by an active whole-class discussion about where the entropy comes from during an irreversible process.
  • Theoretical Mechanics (4 years)
      1. Which pairs of events (if any) are simultaneous in the unprimed frame?

      2. Which pairs of events (if any) are simultaneous in the primed frame?

      3. Which pairs of events (if any) are colocated in the unprimed frame?

      4. Which pairs of events (if any) are colocated in the primed frame?

    1. For each of the figures, answer the following questions:
      1. Which event occurs first in the unprimed frame?

      2. Which event occurs first in the primed frame?