Midterm PHYS 150 X – Fall 2009 Name ______KEY__________
In this experiment, Chuck and Dean are driving a 40
foot super mini-bus at the usual speed of v = 0.6 ft/ns with a shrinkage
factor of 
= 0.8 = 
. Chuck,
as usual, is in the rear of the mini-bus and is adjacent to Anne at the start
of the experiment when x = X = 0 and t = T = 0. Dean is in the front of the mini-bus. The
spacetime diagram has labeled x and t axes.
Remember that lowercase letters are Earth coordinates and uppercase
ones are bus coordinates.
1.
Draw the X and T-axes on the graph.
2.
Use the scale of the x and t axes to
calibrate the X and T-axes.
3.
Locate and label Chuck’s worldline.
4.
Use Dean’s spacetime location at T = 0
to draw his worldline.
5.
According to A & B, how long is
the mini-bus? 32 feet
6.
Is this consistent with the
intersection of Dean’s worldline with the x-axis.
Yes X
When Chuck=s
clock reads zero, he releases a robotic pigeon which flies to Dean. According to Chuck and Dean the pigeon is
zipping along at 2
the speed of light, 0.5 ft/ns.
7.
Label as point A, the starting point
for this pigeon’s trip in the mini-bus.
8.
Chuck and Dean assign what coordinates
to the spacetime point where Dean catches the pigeon? XB = 40 feet and TB = 80 ns. Label that point B on the spacetime diagram.
9.
Use points A and B to draw a dashed
line representing the worldline of the pigeon as it travels from Chuck to Dean.
At the same instant, according to Chuck and Dean,
that Chuck released his pigeon; Dean released a pigeon which flies to Chuck
with a speed of 
ft/ns.
Note that Dean’s pigeon flies at a different speed than Chuck’s
pigeon!
10. Find
the starting point for the pigeon=s
trip from Dean to Chuck and label it point C.
11. C
& D assign what coordinates to the spacetime point D where Chuck catches
the pigeon?
XD = 0 feet and TD = 120 ns. Label
that point D on the spacetime diagram.
12. Draw
a dashed line representing the worldline of the pigeon as it travels from Dean
to Chuck.
The velocity of an object in any reference frame
is just the distance traveled divided by the time of flight in that frame. In the Earth frame v = 
and in the bus frame V = 
.
13. Use
the graph to find the coordinates Anne and Bev assign to point A.
xA = 0 ft, tA = 0 ns.
14. To
point B.
xB = 110 ft,
tB = 130
ns.
15. To
point C.
xC = 50 ft ,
tC = 30
ns.
16. To
point D.
xD = 90 ft ,
tD = 150
ns.
Based on the
coordinates listed above, what is the velocity of the robotic pigeon according
to A & B? (Leave the answers as fractions.)
17. Velocity
of the pigeon traveling from Chuck to Dean = 110/130 = 11/13 ft/ns
18. Velocity
of the pigeon traveling from Dean to Chuck = (90 – 50)/(150-30) = 40/120 = 1/3 ft/ns
The addition of velocities equation, w = (v +
U)/(1 + Uv/c2), can also be used to find the velocity of the pigeon
according to A & B. Remember,
that since the velocities are all given as fractions of the speed of light, c
is equal to 1 in the velocity addition equation.
w = velocity of the pigeon with respect to A &
B
v = velocity of the mini-bus with respect to A
& B
U = velocity of the pigeon with respect to the bus
Use the velocity addition equation to find the
velocity of the pigeon according to A & B.
Remember velocity is direction dependent: velocities toward the right are positive
while velocities towards the left are negative.
19. Velocity
of the pigeon traveling from Chuck to Dean = 
= 
= 
20. Velocity
of the pigeon traveling from Dean to Chuck = 
= 
= 
= 
21. The
velocities found in questions 15) and 16) ought to agree with those found in
17) and 18). If they do not, explain the
reasons for the discrepancies. Yipee, they agree!!
22. Label
the spacetime point were the pigeons pass one another as E.
Use the spacetime graph to answer questions 23)
and 24).
23. What
coordinates do Anne and Bev assign to point E?
xE = 65 feet and tE = 75 ns
24. What
coordinates do Chuck and Dean assign to point E?
XE = 24 feet and TE = 46 ns
Check your answers to 23) and 24) by using the
Lorentz equations:
X =
and T = 
25. XE
from the Lorentz equation = 25 feet
TE from the Lorentz equation
= 45 ns
The agreement is better than I expected!
