Suppose a spaceship heading straight toward the Earth at \(0.750 c\) can shoot a canister at \(0.500 c\) relative to the ship. (a) What is the velocity of the canister relative to Earth, if it is shot directly at Earth? (b) If it is shot directly away from Earth?

In trying to understand the universe, one of the monumental leaps in physics was the development of the theory of special relativity by Albert Einstein in 1905. This theory revolutionized our understanding of time, space, and motion.

Special relativity introduced two fundamental concepts: the laws of physics are the same in all inertial frames of reference, and the speed of light in a vacuum is constant and absolute, regardless of the observer's motion or the source of light.

One of the interesting outcomes of this theory is time dilation, in which time can appear to move slower for an object in motion relative to a stationary observer. Another outcome is length contraction, where the length of an object moving at a significant fraction of the speed of light will appear to contract along the direction of motion to a stationary observer. Both of these are deeply connected to the idea that speed of light remains unchanged, which brings us to the need for relativistic velocity addition when dealing with objects moving at high velocities.

The speed of light in a vacuum is one of the most important constants in physics, denoted by the symbol 'c'. It is precisely 299,792,458 meters per second. No matter how fast you are moving when you measure the speed of light, you will always get the same value—a postulate that stands at the core of special relativity.

Since nothing can travel faster than light, this leads to fascinating implications when it comes to measuring velocities of objects moving at speeds comparable to this cosmic speed limit. At velocities that are a significant fraction of the speed of light, adding speeds does not follow the simple arithmetic we use at lower velocities—hence the introduction of relativistic velocity addition.

Relative velocity is how fast an object is moving in respect to another frame of reference. In our everyday experiences at low speeds, when two objects are moving in the same direction, we subtract their speeds to determine the relative velocity, and when they're moving toward each other, we add their speeds. But this intuition fails at relativistic speeds—such as in the case of a spacecraft moving at a substantial fraction of the speed of light.

In the provided textbook exercise, the velocities involved require the use of the relativistic velocity addition formula, due to the high speeds (significant fractions of the speed of light). The formula \( v = \frac{{v_1 + v_2}}{{1 + \frac{{(v_1)(v_2)}}{c^2}}} \) accounts for the fact that as velocities approach the speed of light, their combination behaves non-intuitively: they don't simply add up.

Using this formula, when the spaceship shoots a canister toward Earth at \(0.750 c\) relative to the spaceship, and when the canister has a velocity of \(0.500 c\) relative to the ship, the velocity of the canister relative to Earth is computed differently when it is shot directly at Earth compared to when it is shot directly away from Earth. As it's a bit counterintuitive, it emphasizes the importance of understanding relative velocity in the context of special relativity.