Problem 39 - Elementary Algebra

Question

According to Einstein's theory of relativity, two velocities \(v_{1}\) and \(v_{2}\) are not added according to \(v=v_{1}+v_{2},\) but rather by \(v=\frac{v_{1}+v_{2}}{1+\frac{v_{1} v_{2}}{c^{2}}}\) Write this complex fraction as a simple fraction. Einstein's formula is really only applicale for velocities near the speed of light \((c=186,000\) miles persecond). At very much lower velocities, such as 500 miles per hour, the formula \(v=v_{1}+v_{2}\) provides an extremely good approximation.

Step-by-Step Solution

Answer

Question: Explain how Einstein's formula for adding velocities simplifies at low velocities and why it provides a good approximation for the standard formula. Answer: At low velocities, much smaller than the speed of light (c), the term \(v_{1}v_{2}/c^{2}\) is insignificant compared to \(c^2\). In this case, the denominator of Einstein's formula can be approximated as simply \(c^2\). Consequently, Einstein's formula becomes \(v \approx (v_{1}+v_{2})\cdot\frac{c^{2}}{c^{2}}=v_{1}+v_{2}\), which is the standard formula for adding velocities. This simplification shows that at low velocities, Einstein's formula provides an extremely good approximation for the standard formula.

Step 1: Simplify Einstein's formula for adding velocities

Einstein's formula for adding two velocities \(v_1\) and \(v_2\) is given by \(v=\frac{v_{1}+v_{2}}{1+\frac{v_{1} v_{2}}{c^{2}}}\). Let's simplify this fraction:

Step 2: Simplify the numerator

The numerator is already in its simplest form: \(v_{1}+v_{2}\).

Step 3: Simplify the denominator

The denominator is \(1+\frac{v_{1} v_{2}}{c^{2}}\). To make it a simple fraction, we need to find the least common denominator (LCD) and combine the terms. The LCD is \(c^2\): $$ 1+\frac{v_{1} v_{2}}{c^{2}} = \frac{c^{2}+v_{1} v_{2}}{c^{2}} $$

Step 4: Rewrite Einstein's formula with the simplified denominator

Now that the denominator is simplified, we can plug it back into the formula for \(v\): $$ v = \frac{v_{1}+v_{2}}{\frac{c^{2}+v_{1} v_{2}}{c^{2}}} $$

Step 5: Simplify the fraction

Now we can multiply the numerator and the denominator of the fraction by the inverse of the denominator: $$ v = (v_{1}+v_{2})\cdot\frac{c^{2}}{c^{2}+v_{1} v_{2}} $$

Step 6: Compare Einstein's formula to the standard formula at low velocities

At low velocities (much smaller than the speed of light \(c\)), the term \(v_{1}v_{2}/c^{2}\) is insignificant as compared to \(c^2\). This means that the denominator \(c^{2}+v_{1} v_{2}\) can be approximated as simply \(c^2\). In this case, Einstein's formula becomes: $$ v \approx (v_{1}+v_{2})\cdot\frac{c^{2}}{c^{2}}=v_{1}+v_{2} $$ Thus, at low velocities, Einstein's formula reduces to the standard formula for adding velocities (\(v = v_1+v_2\)), providing an extremely good approximation.

Key Concepts

Velocity Addition

When we think of adding velocities, our everyday experience might suggest we simply combine the numbers, much like adding two distances. If one car is moving at 40 mph and another at 60 mph, we naturally think their combined speeds would be 100 mph. However, this intuitive approach to velocity addition does not hold true when we enter the realm of high speeds, particularly those near the speed of light.
In the context of Einstein's theory of Special Relativity, velocity addition is not straightforward. If you are on a spaceship traveling at speed \(v_1\) and you observe another ship speeding by at \(v_2\), their velocity relative to you isn't simply \(v_1 + v_2\), but rather it is calculated using the formula: \(v = \frac{v_{1} + v_{2}}{1 + \frac{v_{1} v_{2}}{c^{2}}}\). This formula ensures that no object's velocity exceeds the speed of light \(c\), which is the cosmic speed limit according to relativity.
Einstein's velocity addition is strikingly different from what we experience in our everyday lives, and it becomes essential when dealing with particles moving at a significant fraction of the speed of light. The algebraic simplification of this formula, as shown in the exercise, helps in understanding these complex interactions in a more tangible way.

Special Relativity

Albert Einstein's Theory of Special Relativity revolutionized our understanding of space, time, and how they relate to each other. One of the theory's core principles is that the laws of physics are the same for all observers in uniform motion relative to one another. It implies that the speed of light in a vacuum is constant (\(c\)) and is the same for all observers, regardless of their relative movement.
Special relativity has profound implications, particularly the idea that time can dilate and lengths can contract based on the relative speeds of objects. For instance, time ticks slower for someone moving close to the speed of light compared to someone at rest. This mind-bending concept has been confirmed by numerous experiments and is essential in the functioning of technologies like GPS.
Specific to velocity addition, special relativity teaches us that speeds don't just add up as they would in a normal, 'classical' context. Instead, they combine in a way that respects the unchanging speed of light, leading to the special velocity addition formula we see in Einstein's framework. Its implications are extraordinary, showing that our everyday perceptions are but a shadow of the broader, stranger reality revealed by high-speed physics.

Algebraic Simplification

Algebraic simplification is a process we use to make complex algebraic expressions more manageable. It involves operations like combining like terms, finding least common denominators, and canceling factors to simplify expressions into an equivalent, but simpler form. This process is crucial in solving equations, manipulating formulas, and understanding algebraic concepts in a clearer way.
The simplification of Einstein's velocity addition formula exemplifies the need for algebraic dexterity. Within this framework, simplifying the denominator by finding a common denominator and consolidating terms can make the difference between a complex fraction that is difficult to interpret and a more straightforward expression. As the original exercise demonstrates, multiplying both the numerator and denominator by the inverse of the fraction's denominator converts the complex fraction into a simple one that can be easily compared with classical velocity addition.
Understanding algebraic simplification is not only key to mastering high school mathematics but is also essential to explore advanced topics like relativity. By simplifying complicated equations, we can unveil relationships and insights that may initially be hidden within the complexity of the original expressions.