Question

Ravi and Swati are twins and they are being separated at a rate of 0.80 c. Ravi and Swati each send out a radio signal once a year while Ravi is away. How many signals does Ravi receive for a trip of 15 years?

• A. 3 signals
• B. 5 signals
• C. 9 signals
• D. No signal

Hint:

This is a classic twin paradox-style problem. The key is recognizing that the rate of receiving signals is altered by the relative motion. For objects moving apart, the time between received signals is longer than the time between sent signals.

Answer 1

The Correct Option is B

Step 1: Identify the physical principle. Since the source of the signals (Swati) is moving away from the receiver (Ravi) at a relativistic speed, we must use the relativistic Doppler effect to find the time interval between the reception of signals.
Step 2: Recall the formula for the relativistic Doppler effect (time interval). The time interval between received signals (\(T_{\text{received}}\)) is related to the time interval between sent signals (\(T_{\text{sent}}\)) by the formula: \[ T_{\text{received}} = T_{\text{sent}} \sqrt{\frac{1 + v/c}{1 - v/c}} \] where \(v\) is the relative velocity of separation.
Step 3: Substitute the given values. - Time interval between sent signals, \(T_{\text{sent}} = 1\) year. - Relative velocity, \(v = 0.80c\). \[ T_{\text{received}} = (1 \, \text{year}) \sqrt{\frac{1 + 0.80}{1 - 0.80}} = \sqrt{\frac{1.8}{0.2}} = \sqrt{9} = 3 \, \text{years} \] This means Ravi receives a signal from Swati every 3 years.
Step 4: Calculate the total number of signals received. The trip lasts for 15 years from Ravi's perspective. Since he receives one signal every 3 years, the total number of signals received is: \[ \text{Number of signals} = \frac{\text{Total trip time}}{T_{\text{received}}} = \frac{15 \, \text{years}}{3 \, \text{years/signal}} = 5 \, \text{signals} \]