Question

A rocket ship approaching Earth at \(0.90 c\) fires a missile toward Earth with a speed of \(0.50 c,\) relative to the rocket ship. As viewed from Earth, how fast is the missile approaching Earth?

Short answer

Answer: The total speed of the missile as viewed from Earth is approximately 0.9655c.

Step-by-step solution

Write down the given values

We are given the speed of the rocket ship (\(u\)) as \(0.90c\) and the speed of the missile relative to the rocket ship (\(v'\)) as \(0.50c\). Our goal is to find the speed of the missile as seen from Earth (\(v\)).

Use the relativistic velocity addition formula

Now, plug in the given values into the relativistic velocity addition formula: $$v = \frac{u + v'}{1 + \frac{u \cdot v'}{c^2}}$$

Calculate the speed of the missile as seen from Earth

By substituting the given values into the formula, we have: $$v = \frac{(0.90c) + (0.50c)}{1 + \frac{(0.90c) \cdot (0.50c)}{c^2}}$$ To simplify this expression, we can divide both the numerator and denominator by \(c\): $$v = \frac{0.90 + 0.50}{1 + \frac{0.90 \cdot 0.50}{c}} = \frac{1.40}{1 + \frac{0.45}{c}}$$ Then, use the fact that \(c\) is a constant (the speed of light) and note that the denominator becomes unitless: $$v = \frac{1.40}{1 + 0.45} = \frac{1.40}{1.45}c$$ After dividing the numerator by the denominator, we obtain: $$v = \frac{1.40}{1.45}c = 0.9655c$$ Thus, as viewed from Earth, the missile is approaching Earth at a speed of approximately \(0.9655c\).

Key concepts

Special Relativity

Imagine you are aboard a spaceship, zooming through the cosmos close to the speed of light. As counterintuitive as it seems, the universe changes rules when you travel that fast. This is where Albert Einstein's theory of special relativity comes into play. It provides a new framework for looking at the physics of objects moving at a significant fraction of the speed of light, known as relativistic speeds.

One of the most groundbreaking ideas of special relativity is that the laws of physics are the same for all observers, regardless of their constant velocity. This means that whether you're sitting still on Earth or flying through space, the same physical laws apply. However, special relativity introduces modifications to more classical concepts like time, distance, mass, and energy for objects in high-speed motion. Most notably, it tells us that as an object's speed approaches the speed of light, significant changes occur to these quantities that classical physics can't account for.

An essential application of special relativity is in the equation of relativistic velocity addition. This concept is crucial when determining how fast an object appears to be moving relative to another object when both are moving at speeds close to the speed of light.

Speed of Light

The speed of light, denoted by the symbol 'c,' is more than just a constant—it is the cosmic speed limit, clocking in at approximately 299,792,458 meters per second (in a vacuum). This 'c' plays a pivotal role in the theories of relativity and in understanding the fabric of space-time.

One of Einstein's postulates in special relativity is that the speed of light in a vacuum is the same for all observers, regardless of their motion or the motion of the light source. In other words, no matter how fast you're moving, you'll measure the same speed of light as someone who's standing still. This invariant speed has profound implications, for it leads to effects such as time dilation and length contraction for objects in rapid motion. It acts as a fundamental constant in equations such as E=mc^2, which shows the equivalence of mass and energy, and in the relativistic velocity addition formula.

Velocity Calculation in Physics

Velocity calculation in classical physics is quite simple: if you want to compute the speed at which two objects approach each other, you just add up their velocities. If a car traveling north at 60 km/h passes another car going south at 70 km/h, their relative speed is 130 km/h. But, as the speeds involved get close to the speed of light, this simple addition no longer gives the correct result. Relativistic effects come into play.

The relativistic velocity addition formula becomes necessary when dealing with velocities that are a significant fraction of the speed of light. As seen in the exercise, when a rocket ship travels at 0.90c and fires a missile at 0.50c relative to itself, the velocities cannot be simply added up. Instead, the formula:
\[\begin{equation}v = \frac{u + v'}{1 + \frac{u \cdot v'}{c^2}}\end{equation}\] must be used where 'u' and 'v' are the velocities of the objects and 'c' is the constant speed of light. This equation ensures that the resulting speed does not exceed the speed of light and adheres to the principles of special relativity.

To provide a clearer image for the solution from the exercise, we observed that even when the rocket ship and the missile are both moving at high speeds, their resulting velocity, as observed from Earth, does not merely equal their arithmetic sum but instead is calculated by this more complex formula, resulting in a value of approximately 0.9655c.