An observer measures the length of an object. In which of the following cases is the measured length at a minimum?
Choose one answer.
a. The observer and the object move with the same speed v in the same direction.
b. The observer and the object move with the same speed v in the opposite direction.
c. The observer and the object move with the same speed v in perpendicular directions.
d. The observer moves at an angle 30° to the direction of the object, but both move with the same speed.
Case (b), which has the highest relative velocity (2v), and the greatest relativistic length contraction.
To understand which case would result in the minimum measured length, we need to consider the phenomenon of length contraction in special relativity.
According to Einstein's theory of special relativity, when an object moves with a significant fraction of the speed of light, its length appears contracted or shortened when observed from a relatively stationary frame of reference. This phenomenon is known as length contraction.
Now let's analyze each of the given cases to determine the one in which the measured length is at a minimum:
a. The observer and the object move with the same speed v in the same direction:
In this case, the observer and the object are both moving in the same direction at the same speed. From the observer's frame of reference, the relative speed between them is zero. Therefore, there would be no length contraction observed, and the measured length would be the actual (uncontracted) length of the object.
b. The observer and the object move with the same speed v in the opposite direction:
In this scenario, the observer and the object are moving towards each other at the same speed. From the observer's frame of reference, the relative speed between them would be the sum of their speeds, resulting in a non-zero value. This relative velocity would cause length contraction, and therefore, the measured length of the object would be smaller than its actual length.
c. The observer and the object move with the same speed v in perpendicular directions:
In this case, the observer and the object are moving perpendicular to each other at the same speed. From the observer's frame of reference, the relative speed between them is still zero in the direction parallel to the object's motion. Therefore, there would be no length contraction observed, and the measured length would be the actual length of the object.
d. The observer moves at an angle 30° to the direction of the object, but both move with the same speed:
In this scenario, the observer is moving at an angle to the direction of the object's motion. The component of the observer's velocity that is parallel to the object's motion would contribute to the relative velocity measured by the observer. Since the relative velocity is non-zero, length contraction would occur, causing the measured length of the object to be smaller than its actual length.
Based on the above analysis, option b. is the correct answer. When the observer and the object move with the same speed v in the opposite direction, the observed length would be at a minimum due to length contraction.