Question

9\. How many successive velocity increments of \(\frac{1}{2} c\) from the instantaneous rest frame are needed to produce a resultant velocity of (i) \(0.99 c\), (ii) \(0.999 c\) ? [Answer. 5,7 .

Short answer

Answer: (i) It takes 5 velocity increments to achieve \(0.99c\). (ii) It takes 7 velocity increments to achieve \(0.999c\).

Step-by-step solution

Define the problem and identify the variables

In this problem, we have the following variables: - Each velocity increment (\(\frac{1}{2} c\)) - Desired velocities: (i) \(0.99c\) and (ii) \(0.999c\) - Speed of light (\(c\)) We are to find the number of increments required to reach the desired velocities.

Apply the relativistic velocity addition formula iteratively for case (i)

For the first case, we want to achieve a resultant velocity of \(0.99c\). We will apply the formula repeatedly until we reach this velocity: 1. Starting with the instantaneous rest frame, the velocity is 0. 2. Add the first increment of \(\frac{1}{2}c\). Using the formula with \(u = 0\) and \(v = \frac{1}{2}c\), we have: $$\frac{0 + \frac{1}{2}c}{1 + \frac{0 \cdot \frac{1}{2}c}{c^2}} = \frac{\frac{1}{2}c}{1} = \frac{1}{2}c$$ 3. Add the second increment: $$\frac{\frac{1}{2}c + \frac{1}{2}c}{1 + \frac{\frac{1}{2}c \cdot \frac{1}{2}c}{c^2}} = \frac{c}{1 + \frac{1}{4}} = \frac{4}{5}c \approx 0.8c$$ 4. Add the third increment: $$\frac{\frac{4}{5}c + \frac{1}{2}c}{1 + \frac{\frac{4}{5}c \cdot \frac{1}{2}c}{c^2}} \approx \frac{0.9c}{1 + 0.45} \approx \frac{9}{14}c \approx 0.9643c$$ 5. Add the fourth increment: $$\frac{\frac{9}{14}c + \frac{1}{2}c}{1 + \frac{\frac{9}{14}c \cdot \frac{1}{2}c}{c^2}} \approx \frac{0.9857c}{1 + 0.32} \approx 0.99c$$ So, it takes 5 increments to achieve a resultant velocity of \(0.99c\).

Apply the relativistic velocity addition formula iteratively for case (ii)

Now, we will repeat the process for the second case, aiming for a resultant velocity of \(0.999c\): 1. We already know that after 4 increments, we are at \(\approx 0.99c\). 2. Add the fifth increment: $$\frac{0.99c + \frac{1}{2}c}{1 + \frac{0.99c \cdot \frac{1}{2}c}{c^2}} \approx \frac{0.9950c}{1 + 0.4950} \approx \frac{995}{1990}c \approx 0.998c$$ 3. Add the sixth increment: $$\frac{\frac{995}{1990}c + \frac{1}{2}c}{1 + \frac{\frac{995}{1990}c \cdot \frac{1}{2}c}{c^2}} \approx \frac{0.99874c}{1 + 0.2493} \approx \frac{99874}{99925}c \approx 0.99949c$$ 4. Add the seventh increment: $$\frac{\frac{99874}{99925}c + \frac{1}{2}c}{1 + \frac{\frac{99874}{99925}c \cdot \frac{1}{2}c}{c^2}} \approx \frac{0.9997475c}{1 + 0.12487} \approx 0.999c$$ It takes 7 increments to achieve a resultant velocity of \(0.999c\).

Final Answer: Number of increments

For the given cases, the number of successive velocity increments required are: (i) 5 increments to reach \(0.99c\) (ii) 7 increments to reach \(0.999c\).

Key concepts

Velocity Increments

In relativistic physics, when objects move at speeds comparable to the speed of light, adding velocities isn't as straightforward as in classical physics. The concept of velocity increments helps us understand this complex world.

• A velocity increment is a small addition to an object's speed.
  It's applied iteratively to gradually increase the object's speed.
• In our exercise, the given velocity increment is \(\frac{1}{2} c\).
  Here, \(c\) represents the speed of light.

Each increment uses the relativistic velocity addition formula, consistently adjusting the object's speed until it nears our target speed of a significant fraction of \(c\).
This approach captures the essence of building up speed gradually, within the limits of relativity.

Speed of Light

The speed of light, denoted as \(c\), is the maximum speed at which all energy, matter, and information in the universe can travel.

• It is approximately \(3 \times 10^8\) meters per second.
• In relativistic physics, \(c\) is not just a number; it acts as a fundamental limit.

Why is the speed of light so crucial in physics?
As objects approach \(c\), their mass effectively increases and they require more energy to continue accelerating.
This makes \(c\) an unreachable speed for objects with mass, influencing how we calculate velocity increments in relativistic scenarios.

Instantaneous Rest Frame

The concept of the instantaneous rest frame is pivotal for understanding relativity.
It refers to the frame of reference where an object, at a particular moment, is at rest or has no velocity.

• This frame changes as the object's velocity increases.
  Each time we apply a velocity increment, the rest frame is adjusted to that moment.
• Relativistic calculations happen more easily in the object's instantaneous rest frame.

Utilizing this frame, we can incrementally apply \(\frac{1}{2} c\) at each step without directly referencing influences from other frames.
Thus simplifying the calculations while respecting relativistic laws.

Resultant Velocity

In the context of our exercise, the resultant velocity is the final speed achieved after several applications of velocity increments.
It answers the core question: "What will be the speed after applying `x` increments?"

• This velocity takes into account all the relativistic corrections from each increment.
  The successive calculations show the balancing act between applied increments and relativistic limits.
• For instance, achieving a velocity of \(0.99c\) requires 5 increments, while \(0.999c\) requires 7.
  It demonstrates the nonlinear nature of speed increase as it approaches \(c\).

Thus, the resultant velocity emerges as challenge wrangling with the very fabric of space and time, encapsulated in calculations bound by the speed of light.