When observed from the sun at a particular instant, Earth and Mars appear to move in opposite directions with speeds \(108,000 \mathrm{km} / \mathrm{h}\) and \(86,871 \mathrm{km} / \mathrm{h}\), respectively. What is the speed of Mars at this instant when observed from Earth?

Understanding the motion of objects moving in opposite directions is crucial when studying relative velocity. Consider two vehicles traveling on a highway; if they are moving in opposite directions, their relative speed increases because they are moving away from each other.

Now, let's apply this concept to celestial bodies such as planets. When Earth and Mars move in opposing directions as seen from the sun, we intuitively understand that someone on Earth would perceive Mars as moving faster because it is not just moving, but also because Earth is moving in the opposite direction. This visual scenario is foundational for calculating the relative velocity of two objects in space. The step-by-step solution provided illustrates this concept by assigning opposite signs to their velocities and using the fact that Mars' velocity would be negative if Earth's is taken to be positive.

A scalar quantity is defined by its magnitude and has no direction. This characteristic contrasts with vector quantities, which have both magnitude and direction. Common examples of scalar quantities include temperature, mass, and speed.

In our exercise example, we are interested in the speed of Mars relative to Earth. Even though the initial calculation yields a negative result, this negative sign merely indicates direction and not the magnitude of speed. Since speed is a scalar quantity, we take the absolute value of the calculated result to express the speed of Mars with respect to Earth without regard for the direction, which results in an answer of 194,871 km/h. When discussing scalar quantities, always remember that they are independent of direction; they simply convey size or amount.

When examining the velocity of planets, it's important to consider that their motion is relative to an observational reference point. In our textbook problem, the velocities of Earth and Mars are initially given with respect to the sun. However, if we want to calculate Mars' velocity from the perspective of an observer on Earth, we need to apply the concept of relative velocity.

The formula to find the velocity of one object relative to another is crucial here. It defines how to combine the velocities of Earth and Mars to determine how fast Mars appears to be moving from Earth's viewpoint. This calculation is especially intriguing because it allows us to step into a frame of reference that is in motion itself—the moving Earth—offering a dynamic perspective on how we measure motion in our solar system.