Before we present the first example of solving a problem in relativity, we state a strategy you can use as a guideline for these calculations.
Problem-solving strategy: relativity
Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity
v .
Identify exactly what needs to be determined in the problem (identify the unknowns).
Make certain you understand the conceptual aspects of the problem before making any calculations (express the answer as an equation). Decide, for example, which observer sees time dilated or length contracted before working with the equations or using them to carry out the calculation. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
Determine the primary type of calculation to be done to find the unknowns identified above (do the calculation). You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.
Note
that you should not round off during the calculation . As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem solution, but do not use a rounded number in a subsequent calculation. Also, check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, which has few everyday examples to provide experience with what is reasonable. But you can look for velocities greater than
c or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).
Time dilation in a high-speed vehicle
The Hypersonic Technology Vehicle 2 (HTV-2) is an experimental rocket vehicle capable of traveling at 21,000 km/h (5830 m/s). If an electronic clock in the HTV-2 measures a time interval of exactly 1-s duration, what would observers on Earth measure the time interval to be?
Strategy
Apply the time dilation formula to relate the proper time interval of the signal in HTV-2 to the time interval measured on the ground.
Solution
Identify the knowns:
Identify the unknown:
Express the answer as an equation:
Do the calculation. Use the expression for
to determine
from
:
Significance
The very high speed of the HTV-2 is still only 10
-5 times the speed of light. Relativistic effects for the HTV-2 are negligible for almost all purposes, but are not zero.
How fast must a vehicle travel for 1 second of time measured on a passenger’s watch in the vehicle to differ by 1% for an observer measuring it from the ground outside?
Strategy
Use the time dilation formula to find
v/c for the given ratio of times.
Solution
Identify the known:
Identify the unknown:
v/c .
Express the answer as an equation:
Do the calculation:
Significance
The result shows that an object must travel at very roughly 10% of the speed of light for its motion to produce significant relativistic time dilation effects.
