The proper length

Problem:

A car of rest length 5 m passes through a garage of rest length 4 m.  Due to Lorentz contraction, the car is only 3 m long in the garage's rest frame.  Find the length of the garage in the car's rest frame.

Solution:

Problem:

The radius of the galaxy is 3*1020m, measured in its own rest frame.
(a)  If the time it takes a spaceship to cross the entire galaxy is 300 years measured in the spaceship's rest frame, what is the relative speed of spaceship and galaxy?
(b)  How much time elapses on Earth during this trip?

Solution:

Problem:

A space ship has a proper length of 100 m.  It travels close to the Earth's surface with a constant speed of 0.8 c.  Earth observers decide to measure the length of the ship by erecting two towers that coincide with the ends of the ship simultaneously (in the Earth's frame) as it passes by.
(a)  How far apart do the observers on Earth build the towers?
(b)  How long do the observers on Earth say it takes for the nose of the ship to travel from tower A to tower B?
(c)  How long, according to the measurements in the spaceship frame, does it take for the nose of the ship to travel from tower A to tower B?
(d)  As measured by passengers in the spaceship, how far apart are the two towers?
(e)  In the spaceship frame, how long does it take a beam of light to travel from the front to the rear of the spaceship?
(f)  How much time, according to the observers on Earth, is required for a beam of light to travel from the front to the rear of the moving spaceship?
(Give numerical answers!)

 Solution:

Problem:

Assuming that the rest radius of earth is 6,400 km and its orbital speed about the sun is 30 km/s, how much does earth's diameter appear to be shortened along its direction of motion to an observer on the sun?

Solution:

Problem:

A space ship has a proper length of 100 m.  It travels close to the Earth's surface with a constant speed of 0.8 c.  Earth observers decide to measure the length of the ship by erecting two towers that coincide with the ends of the ship simultaneously (in the Earth's frame) as it passes by.
(a)  How far apart do the observers on Earth build the towers?
(b)  How long do the observers on Earth say it takes for the nose of the ship to travel from tower A to tower B?
(c)  How long, according to the measurements in the spaceship frame, does it take for the nose of the ship to travel from tower A to tower B?
(d)  As measured by passengers in the spaceship, how far apart are the two towers?
(e)  In the spaceship frame, how long does it take a beam of light to travel from the front to the rear of the spaceship?
(f)  How much time, according to the observers on Earth, is required for a beam of light to travel from the front to the rear of the moving spaceship?
(Give numerical answers!)

Solution:

Problem:

A rod of length L0 is inclined at angle θ from the x-axis in its rest frame.  Find the inclination angle of the rod as measured by an observer moving with relativistic speed “v” in the x-direction.

Solution:

Problem:

A rectangular plate of dimensions a × b moves at relativistic velocity V = Vi as shown in the figure.  In the rest frame of the rectangle, side a makes an angle θ with respect to the x axis.

image

(a)  Sketch the shape of the plate as measured by an observer in the laboratory frame.  Write down expressions for the lengths of all sides and the values of all interior angles in terms of β = V/c, γ = (1 - β2), and θ .
(b)  Evaluate your expressions for the case θ = π/4 and V = ⅔½c.

Solution:

Problem:

A stick of length l is fixed at an inclination of angle θ from the x1-axis in its own rest frame K.  Consider an observer in a frame K′ moving along the x1-axis with speed v.  What does this observer measure for
(a)  the length of the stick and
(b)  the angle of the stick with respect to the x1'-axis?

Solution: