Physics & Astronomy
The length of the object is contracted along the direction of motion as the object is moving with relativistic speed is called the length contraction. The spacecraft with the shape of sphere is moving with speed 0.5c relative an observer on Earth. So the diameter of the spacecraft is reduced along the direction of spacecraft motion. Hence, the shape of the spacecraft is ellipse instead of spherical shape.
The expression for time interval measured in the moving frame of reference is, Here, is the time interval measured in the rest frame and is the relativistic factor.
The expression for the relativistic factor is, Here, v is the speed of the astronaut and c is the velocity of light.
Rearrange the equation for as follows: Substitute for . The observer in the astronaut's frame measures the proper time and observer in the Earth's frame measures the dilated time.
Substitute for and 0.950 c for v in the expression . Therefore, the proper time interval is .
The observer in the astronaut's frame measures the proper length and observer in the Earth's frame measures the contracted length.
The expression for the distance measured by the astronaut is, Here is the proper length and is the speed of the astronaut.
Substitute for and for . Therefore, the distance to Alpha Centaury measured by the astronaut is .
Suppose the quasar moving with relativistic speed and it is emitting a light. The speed of the light measured by the two observers is same even if they are moving relative to one another. Therefore, two observers always agree on the speed of light and the speed of one observer relative to the other because its magnitude is also same as measured by two observers.