In this "lab" (worksheet), you will explore the geometry of Lorentz transformations,
specifically a "boost" from frame S to S^('), and how this changes the coordinates of time and
space. You will need one full sheet of graph paper per person.
Procedure:
Part A - The Spacetime Diagram
On graph paper, lay out the S frame with axis ct vertical and x horizontal. For ct, create axis
ticks from 0 to 15m in increments of 0.5m. For x, go from 0 to 20m in increments of 0.5m.
In this exercise, frame S^(') moves relative to S with speed v=0.6000c. Determine gamma , then
use the Lorentz Transformation equations to draw lines for the ct^(') and x^(') axes on your
graph. Note that S and S^(') have the same origin at x=x^(')=0 and t=t^(')=0.
After you have the ct^(') and x^(') axes drawn, locate and draw lines for
x^(')=2.5,5.0,7.5,10.0,12.5, and 15.0m. Again, use the Lorentz Transformation equations
to help you do this.
Next, locate and draw lines for ct^(')=10.0,7.5,5.0,2.5,-2.5, and -5.0m.
Now that you have both sets of axes set up, choose two events ( A and B ) that are
simultaneous in frame S. Choose the points so that Delta x>=7m and ct>=4m. Label the
points on your graph.
Use the Lorentz Transformation equations to calculate the S^(') coordinates of these points.
Do your calculated values agree with the grid you drew in steps 3 and 4 ? If not, find your
errors and correct them.
Using Delta x,cDelta t,Delta x^('), and cDelta t^('), calculate Delta s^(2) and Delta s^('2). Is it invariant? If not, again correct your
errors.
According to your grid, which event happened first in S^(') ?
Which is greater, Delta x or Delta x^(') ? How are they related? Explain this regarding proper length.
Now choose two events and B^(') ) that are simultaneous in frame S^('). Choose the points so
that Delta x^(')>=7m and ct^(')!=0. Label the points on your graph.
Use the Lorentz Transformation equations to calculate the S coordinates of these points. Do
you calculated values agree with the S grid? If not, find your errors and correct them.
Using Delta x,cDelta t,Delta x^('), and cDelta t^('), calculate Delta s^(2) and Delta s^('2). Is it invariant? If not, again correct your
errors.
According to your grid, which event happened first in S ?
Which is greater, Delta x or Delta x^(') ? How are they related? Explain this regarding proper length.
Using a different color, draw the path through spacetime (the world line) for a beam of light
emitted from the origin. Verify that both frames measure the speed to be c.
Can anyone help me with this lab?
Project Caliper
Lorentz Transformations - Spacetime
Page 1
In this"lab"(worksheet),you will explore the geometry of Lorentz transformations specifically a "boost" from frame S to S', and how this changes the coordinates of time and space. You will need one full sheet of graph paper per person.
Procedure: Part A-The Spacetime Diagram 1. On graph paper,lay out the S frame with axis ct vertical and x horizontal.For ct,create axis ticks from 0 to 15 m in increments of 0.5 m.For x,go from 0 to 20 m in increments of 0.5 m 2. In this exercise,frame Smoves relative to S with speed v = 0.6000c.Determine y,then use the Lorentz Transformation equations to draw lines for the ct' and x' axes on your graph. Note that S and S' have the same origin at x = x' = 0 and t = t' = 0. 3. After you have the ct' and x' axes drawn,locate and draw lines for x' = 2.5, 5.0, 7.5, 10.0, 12.5, and 15.0 m. Again, use the Lorentz Transformation equations to help you do this. 4. Next, locate and draw lines for ct'=10.0,7.5,5.0,2.5,-2.5,and -5.0 m 5. Now that you have both sets of axes set up,choose two events(A and B)that are simultaneous in frame S.Choose the points so that Ax 7 m and ct 4 m.Label the points on your graph. 6. Use the Lorentz Transformation equations to calculate the S' coordinates of these points Do your calculated values agree with the grid you drew in steps 3 and 4? if not, find your errors and correct them 7. Using Ax, cAt, Ax', and ct', calculate As2 and s'2. Is it invariant? If not, again correct your errors. 8. According to your grid, which event happened first in S'? 9. Which is greater, Ax or Ax'? How are they related? Explain this regarding proper length. 1o.Now choose two events (A' and B') that are simultaneous in frame S'.Choose the points so that x' 7 m and ct' # 0. Label the points on your graph. 11.Use the Lorentz Transformation equations to calculate the S coordinates of these points.Do you calculated values agree with the S grid? if not, find your errors and correct them. 12. Using Ax, ct, Ax', and ct', calculate s2 and s'2. Is it invariant? If not, again correct your errors. 13.According to your grid,which event happened first in S? 14. Which is greater, x or Ax'? How are they related? Explain this regarding proper length. 15. Using a different color, draw the path through spacetime (the world line) for a beam of light emitted from the origin.Verify that both frames measure the speed to be c
