Physics 2301: Midterm Two
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Physics 2301: Midterm Two
Lorentz transformation between frames S and Sz moving at relative speed v along the x–axis:
Relativistic
xz = γ(x _ vt)
yz = y
zz = z
tz = γ(_vx/c2 + t) velocity composition:
x = γ(xz + vtz )
y = yz
z = zz
t = γ(vxz /c2 + tz )
u · v =
γ 三 1/ /1 _ v2 /c2
1. (5 points) In inertial frame S two events are deemed to occur at the same place, but with time separation ∆t = 12 seconds. In inertial frame Sz the same two events are deemed to occur with time separation ∆tz = 13 seconds. What is the relative velocity between S and Sz ?
2. (5 points) If the total relativistic energy of a particle of mass m is three times its rest energy, what is the magnitude of its relativistic momentum?
3. (25 points ) Ball A, of rest mass m, travels down the x axis at speed vA /c = 4/5. An identical ball B is moves at vu /c = _3/5 relative to the ground.
(a) Find the energy and momentum of A and of B in the ground frame.
(b) Find the speed of the center of mass of the AB system (relative to the ground).
(c) Find the speeds of A and B in the center of mass frame, and their energies.
(d) Verify that the 4-length of the total system 4-momentum is the same, whether one computes it using p and E in the CM frame or in the ground frame.
(e) What is the speed of B relative to A?
(f) Use the Lorentz transformation and your knowledge of B’s energy and momentum in the ground frame (part (a)) in to find B’s energy and momentum in A’s frame.
4. (15 points) Including the appropriate γ 3 from relativity, the relativistic harmonic oscillator obeys the
equation of motion
d2
Starting at rest with x(0) = x0 we expect x(t) to be an even function of t, so we look for a series solution of the form
x(t) = x0 + a2 t2 + a4 t4 + . . .
Plug the “ansatz” into the differential equation and find the coefficients a2 and a4 .
5. (40 points) Consider the familiar situation: Alan stands on the ground while Beth is at middle of a train of (proper) length 2L, moving with speed v = (4/5)c relative to Alan. Beth’s buddy Becky stands at the front of the train, a distance L away from Beth, and her bestie Bree rides at the back at xz = _L. As Alan and Beth meet (an event they all agree to call their origin event), they have a brief conversation. Beth explains that her intercom radio is broken, and asks Alan to radio messages to her buddies, asking them to walk to the middle of the train. Alan complies, and immediately emits radio signals in both directions.
(a) By considering how long the radio wave will take to catch up to its moving target, find Alan’s ground-based t1 and x1 coordinates for event 1, “radio signal arrives at Becky.”
(b) As it happens, Becky is a fast walker. Upon receipt of the message she immediately starts walking
back at u = _(3/5)c relative to the train. What is her speed relative to Alan as she does so?
(c) By considering their relative speeds in Alan’s frame, find how long it takes Becky to walk to Beth. Add this time to t1 to find the time coordinate t2 of event 2, “Becky arrives at Beth.”
(d) Same questions concerning Bree: what are Alan’s coordinate t3 and x3 for event 3, “message arrives at Bree”?
(e) And if Bree walks at u = +(3/5)c relative to the train, what is her speed relative to Alan?
(f) Given the relative speeds, how long (still working in Alan’s frame) does it take Bree to catch up to Beth? Find Alan’s coordinate t4 for event 4, “Bree arrives at Beth”. According to Alan, which buddy arrives at Beth first?
(g) Find the spacetime interval between events 1 and 3, using Alan’s coordinates for the events. Is it spacelike, timelike or lightlike?
(h) Use the Lorentz transform to find Beth’s labelling of the events 1 and 3, (tz 1 , xz 1 ) and (tz 3 , xz 3 ).
(i) Using Beth’s coordinates, find the interval between events 1 and 3.
(j) Draw spacetime diagrams for Alan and again for Beth, indicating the placement of the events 1-4 and worldlines for the four people.
6. (10 points) A photon heads down the +x axis with given energy E and collides with a 2m at rest, producing a final state with two m’s. One of the m’s flies off at angle relative to the x-axis. Find its energy (E1 in the figure).
2022-04-02