Question

Calculate the velocity, relative to us, in \(\mathrm{m} \mathrm{s}^{-1}\) of an object with a rest mass, \(m_{0},\) of \(1.000 \mathrm{~kg}\) when its mass, \(m,\) is (a) \(1.005 \mathrm{~kg}\), (b) \(1.1 \mathrm{~kg},\) and (c) \(5.0 \mathrm{~kg}\).

Short answer

The velocities for each mass scenario are: (a) Velocity when \(m = 1.005 \mathrm{kg}\) is approximately \(4.24 \times 10^7 \mathrm{m/s}\), (b) Velocity when \(m = 1.1 \mathrm{kg}\) is approximately \(1.40 \times 10^8 \mathrm{m/s}\), and (c) Velocity when \(m = 5.0 \mathrm{kg}\) is approximately \(2.98 \times 10^8 \mathrm{m/s}\).

Step-by-step solution

Understand the relation between rest mass and velocity using the theory of relativity

The theory of relativity shows that as an object moves faster, its mass increases due to relativistic effects. This relationship can be described by the equation: \[m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}\], where \(m\) is the relativistic mass, \(m_0\) is the rest mass of the object, \(v\) is the velocity of the object, and \(c\) is the speed of light in a vacuum (approximately \(3.00 \times 10^8 m/s\)). This formula can be rearranged to solve for velocity.

Solve for velocity

Rearrange the equation to solve for \(v\): \[v = c \sqrt{1 - \left(\frac{m_0}{m}\right)^2}\]. We will use this formula to calculate the velocity for each given value of \(m\).

Calculate velocity when \(m = 1.005 \mathrm{kg}\)

Plug in \(m_0 = 1.000 \mathrm{kg}\) and \(m = 1.005 \mathrm{kg}\) into the equation: \[v = 3.00 \times 10^8 \times \sqrt{1 - \left(\frac{1.000}{1.005}\right)^2}\].

Calculate velocity when \(m = 1.1 \mathrm{kg}\)

Plug in \(m_0 = 1.000 \mathrm{kg}\) and \(m = 1.1 \mathrm{kg}\) into the equation: \[v = 3.00 \times 10^8 \times \sqrt{1 - \left(\frac{1.000}{1.1}\right)^2}\].

Calculate velocity when \(m = 5.0 \mathrm{kg}\)

Plug in \(m_0 = 1.000 \mathrm{kg}\) and \(m = 5.0 \mathrm{kg}\) into the equation: \[v = 3.00 \times 10^8 \times \sqrt{1 - \left(\frac{1.000}{5.0}\right)^2}\].

Perform the calculations

Use a calculator to perform the square root and multiplication for each scenario to find the velocities.

Key concepts

Theory of Relativity

The theory of relativity, primarily developed by Albert Einstein, revolutionized our understanding of space, time, and energy. It consists of two main parts: Special Relativity and General Relativity. Special Relativity, introduced in 1905, addresses how space and time are interwoven and how they are affected by speed. One of its key postulates is that the speed of light in a vacuum is the same for all observers, regardless of their relative motion.

General Relativity, proposed in 1915, extends these concepts to include gravity, describing it as a curvature in spacetime caused by mass and energy. A remarkable prediction of General Relativity is the bending of light around massive objects, something that has been confirmed by astronomical observations.

One of the mind-boggling outcomes of Special Relativity is that as objects move closer to the speed of light, they experience time differently, and their mass appears to increase from the perspective of a stationary observer. This mass increase implies that it would require infinite energy to accelerate an object with mass to the speed of light.

Mass-Velocity Relationship

Intricately linked to the theory of relativity is the mass-velocity relationship, which explains how an object's mass appears to change as its speed approaches the speed of light. This phenomenon is described by the relativistic mass formula:
\[\[\begin{align*} m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}\end{align*}\]\]
In this equation, \(m\) represents the relativistic mass observed, \(m_0\) the rest mass (or the mass of the object when it's not moving relative to the observer), \(v\) is the object's velocity, and \(c\) is the speed of light.

As velocity increases, the denominator of the fraction inside the square root gets smaller, leading to an increase in the relativistic mass. It's important to note that the relativistic mass increase is not an actual change in the intrinsic mass of the object but rather how mass is perceived by an observer moving at a different velocity relative to the object. This concept has profound implications in high-energy physics, such as particle accelerators where particles are sped up to immense speeds.

Special Relativity Equations

To solve problems in special relativity, such as calculating the new velocity of an object when its mass is observed to change, special relativity equations are employed. A key formula derived from Einstein's work is the following:
\[\[\begin{align*} v = c \sqrt{1 - \left(\frac{m_0}{m}\right)^2}\end{align*}\]\]
This specific equation allows us to determine the object's velocity (\(v\)) based on its observed relativistic mass (\(m\)) and rest mass (\(m_0\)), with the constant (\(c\)) being the speed of light.

To effectively utilize this formula, rearrangement and algebraic manipulation are often necessary. Applying these equations requires careful attention to units and a deep understanding that mass does not actually change with speed; instead, it is the inertia—the resistance to acceleration—that increases. This nuance is crucial in understanding relativistic effects without misconceptions.