a) Light takes 5.2 years to travel from a distant planet to earth. If the astronaut travelled from earth to that planet at a speed of 0.8c0.8 \mathrm{c}, how long would it take according to the astronaut’s clock, to reach the planet?
Answer:
To find the time it takes for the astronaut to reach the planet according to the astronaut’s clock, we need to consider time dilation due to special relativity. The time measured by the astronaut, t^(‘)t’, is related to the time measured by an observer on Earth, tt, by the following equation:
t^(‘)=(t)/( gamma)t’ = \frac{t}{\gamma}
where gamma\gamma is the Lorentz factor, given by:
Here, v=0.8 cv = 0.8c is the speed of the astronaut, and cc is the speed of light. The time it takes for light to travel from the planet to Earth is given as 5.2 years, so t=5.2″ years”t = 5.2 \text{ years}.
So, according to the astronaut’s clock, it would take approximately 3.12 years to reach the planet.
b) Calculate the linear momentum, total energy (in MeV\mathrm{MeV} ) and kinetic energy (in MeV\mathrm{MeV} ) of a particle travelling at 0.5c0.5 \mathrm{c}, given that its rest mass is 938MeV938 \mathrm{MeV}.
Answer:
To calculate the linear momentum, total energy, and kinetic energy of a particle traveling at 0.5 c0.5c, where cc is the speed of light, and given that its rest mass is 938MeV938 \mathrm{MeV} (which corresponds to the rest mass of a proton), we can use the following relativistic formulas:
Linear Momentum (pp):
p=gammam_(0)vp = \gamma m_0 v
where m_(0)m_0 is the rest mass, vv is the velocity of the particle, and gamma\gamma is the Lorentz factor, given by:
Therefore, the linear momentum of the particle is approximately 542.09MeV//c542.09 \mathrm{MeV/c}, the total energy is approximately 1082.99MeV1082.99 \mathrm{MeV}, and the kinetic energy is approximately 145.39MeV145.39 \mathrm{MeV}.
c) Two spaceships of proper length L_(0)L_0 approach the earth from opposite direction at velocities +-0.7 c\pm 0.7 c. What is the length of one of the spaceships with respect to the other?
Answer:
To find the length of one of the spaceships with respect to the other, we need to consider the relativistic effect of length contraction. When an object is moving relative to an observer, its length appears contracted along the direction of motion according to the formula:
where LL is the observed length, L_(0)L_0 is the proper length (the length of the object in its rest frame), vv is the relative velocity between the observer and the object, and cc is the speed of light.
In this case, the relative velocity between the two spaceships is the sum of their individual velocities, as they are moving in opposite directions. Therefore, v=0.7 c+0.7 c=1.4 cv = 0.7c + 0.7c = 1.4c. However, the maximum possible relative velocity cannot exceed the speed of light, so we need to use the relativistic velocity addition formula to find the correct relative velocity:
So, the length of one of the spaceships with respect to the other is approximately 0.3421L_(0)0.3421 L_0, meaning it appears to be about 34.21% of its proper length.