(b) (to five digits to show effect)
(b) cannot be less than 1.
(c) Assumption that time is longer in moving ship is unreasonable.
(a) 1.387 km = 1.39 km
(b) 0.433 km
Thus, the distances in parts (a) and (b) are related when .
(a) 4.303 y (to four digits to show any effect)
(b) 0.1434 y
Thus, the two times are related when .
(b) must be ≥1
(c) The Earth-bound observer must measure a shorter length, so it is unreasonable to assume a longer length.
c) (all to sufficient digits to show effects)
(b) Ratio of relativistic to classical momenta equals 1.000000005 (extra digits to show small effects)
(b) The small speed tells us that the mass of a proton is substantially smaller than that of even a tiny amount of macroscopic matter!
The ratio is greater for hydrogen.
(b) The rest mass energy of an electron is 0.511 MeV, so the kinetic energy is approximately 150% of the rest mass energy. The electron should be traveling close to the speed of light.
(a) 1.00 kg
(b) This much mass would be measurable, but probably not observable just by looking because it is 0.01% of the total mass.
The relativistic Doppler effect takes into account the special relativity concept of time dilation and also does not require a medium of propagation to be used as a point of reference (light does not require a medium for propagation).
Relativistic kinetic energy is given as
Classical kinetic energy is given as
At low velocities , a binomial expansion and subsequent approximation of gives:
Substituting in the expression for gives
Hence, relativistic kinetic energy becomes classical kinetic energy when .