A global positioning system (GPS) functions by determining the travel times for EM waves from various satellites to a moving GPS receiver on Earth (car or hiker). If the receiver is to detect a change in the receiver’s position on the order of 3 m, what is the associated change in travel time (in ns) that must be measured?

Verified step by step guidance

1. Understand the relationship between distance, speed, and time. The formula to use here is: $d=vt$, where $d$ is the distance, $v$ is the speed of the electromagnetic (EM) wave (speed of light), and $t$ is the travel time.
2. Rearrange the formula to solve for the change in time: $\mathrm{\Delta t}=\mathrm{\Delta d}/v$, where $\mathrm{\Delta d}$ is the change in distance (3 m in this case) and $v$ is the speed of light ($3\; \times \; 108\mathrm{m/s}$).
3. Substitute the given values into the formula: $\mathrm{\Delta t}=\; 3/\left(3\; \times \; 108\right)$. This will give the change in time in seconds.
4. Convert the result from seconds to nanoseconds (ns). Recall that 1 second is equal to $109$ nanoseconds. Multiply the result from the previous step by $109$ to get the change in time in nanoseconds.
5. Interpret the result. The calculated change in time represents the precision required by the GPS receiver to detect a positional change of 3 meters. This highlights the incredible accuracy of GPS systems in measuring time intervals.

Key Concepts

Speed of Light

The speed of light in a vacuum is approximately 299,792,458 meters per second. This constant is crucial for understanding how electromagnetic (EM) waves, such as those used in GPS, propagate through space. When calculating travel time for signals, knowing this speed allows us to determine how long it takes for a signal to travel a specific distance.

Time of Flight

Time of flight refers to the time it takes for a signal to travel from one point to another. In the context of GPS, it is the duration for an EM wave to travel from a satellite to the receiver. To detect a change in position, the system must measure very small changes in this time, which directly correlates to the distance traveled by the signal.

Precision in Measurement

Precision in measurement is critical in GPS technology, as even minor errors can lead to significant discrepancies in position calculations. To detect a change in position of 3 meters, the system must measure the corresponding change in travel time with high accuracy, which is typically in the nanosecond range. This precision ensures that the GPS can provide reliable location data.