Question

To circle the Earth in low orbit a satellite must have a speed of about \(7.91 \mathrm{~km} / \mathrm{s}\). Suppose that two such satellites orbit the Earth in opposite directions. (a) What is their relative speed as they pass? Evaluate using the classical Galilean velocity transformation equation. (b) What fractional error was made because the (correct) relativistic transformation equation was not used?

Short answer

The relative speed as per the classical Galilean transformation is \(15.82 \, \mathrm{km/s}\). Using the relativistic transformation equation, it is \(15.82000004 \, \mathrm{km/s}\). The fractional error made because the relativistic transformation equation was not used is approximately \(2.5 \times 10^{-9}\).

Step-by-step solution

Use the Galilean Transformation

The classical Galilean velocity transformation equation is simply the addition of velocities. If two objects are each moving at a speed of \(7.91 \, \mathrm{km/s}\) in opposite directions, their relative speed is the sum of these velocities, which equals \(2 \times 7.91 = 15.82 \, \mathrm{km/s}\).

Use the Relativistic Transformation

The relativistic velocity transformation equation, which is derived from Einstein's theory of relativity, would give a different result due to time dilation and length contraction at high speeds. It is given by \(v' = \frac{{v_1 + v_2}}{{1 + (v_1v_2/c^2)}}\). Here, \(v_1 = v_2 = 7.91 \, \mathrm{km/s}\) and \(c\) is the speed of light, which equals \(299792.458 \, \mathrm{km/s}\). By plugging these values in, the relative speed is calculated as \(15.82000004 \, \mathrm{km/s}\).

Calculate the Fractional Error

The fractional error can now be calculated as the difference between the relativistic result and the classical result, divided by the classical result, which equals \(\frac{{15.82000004-15.82}}{{15.82}}\). This results in a fractional error of approximately \(2.5 \times 10^{-9}\). This demonstrates that the classical transformation is a very good approximation at these speeds, but still has a minuscule error due to the nature of relativity.

Key concepts

Relative Speed

When discussing motion, it's common to encounter the concept of relative speed. This refers to the speed of an object as observed from the reference frame of another moving object. Consider two cars traveling towards each other on a highway, each moving at 60 km/h relative to the ground. The relative speed between the two cars is the sum of their speeds, which in this case would be 120 km/h.

Similarly, in the exercise involving two satellites orbiting Earth in opposite directions, the relative speed is the sum of their individual speeds when using classical mechanics, which aligns with the Galilean velocity transformation. This equation assumes that time and space are absolute and not affected by the speeds of the objects, which works well for lower-speed scenarios that we encounter in everyday life. The simplicity of this approach makes it an excellent teaching tool for introducing students to the concepts of motion and relative velocity.

Relativistic Velocity Transformation

High-speed situations, such as those involving objects moving at significant fractions of the speed of light, require a shift from classical to relativistic velocity transformations. These transformations stem from Einstein's theory of special relativity, which takes into account the constancy of the speed of light for all observers and the consequences this has on our perceptions of time and space.

Mathematically, the relativistic transformation for combining velocities is more complex than the Galilean transformation. It incorporates time dilation (time appears to slow down at higher speeds) and length contraction (objects appear shorter in the direction of motion at higher speeds), phenomena that become significant as objects move closer to the speed of light. This complexity shows students that classical mechanics is just an approximation, and a different set of rules begins to apply when dealing with high velocities.

Fractional Error in Classical Mechanics

It's important to understand the limitations of classical mechanics, especially when dealing with higher speeds. Fractional error is a useful concept in this respect as it quantifies how much a classical calculation deviates from a more accurate relativistic one. The fractional error is the difference between two values (in this case, velocities) divided by the more accurate value, typically the one from the relativistic equation.

Students learning about these concepts should recognize that while classical mechanics is simpler and works well for everyday scenarios, it can produce a fractional error when applied to high-speed situations. However, as the exercise demonstrates, when dealing with speeds much lower than the speed of light, such as those of low Earth orbit satellites, the fractional error is minuscule. This reinforces the idea that classical mechanics is still highly relevant in most practical situations.

Low Earth Orbit Satellites

Low Earth orbit (LEO) satellites provide a real-world application of these physics concepts. LEO satellites are typically found at altitudes between 160 to 2,000 kilometers above Earth's surface. Circling the Earth at such altitudes, these satellites have speeds of around 7.91 km/s to maintain their orbit due to gravitational forces and the need to counteract Earth's curvature.

In the context of our exercise, two LEO satellites moving in opposite directions at this speed will encounter each other at a relative speed, which is simply the sum of their speeds according to classical physics. This example helps students link theoretical knowledge with actual objects they can relate to, like satellites that facilitate communications and earth observations. It's a demonstration of how physics principles are used in the design and operation of technologies that are part of our everyday lives.