Problem 68 - University Physics with Modern Physics

Question

At what speed will the length of a meter stick look \(90.0 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

Answer: Approximately \(1.35 \times 10^8 \mathrm{\frac{m}{s}}\).

Step 1: Write down the length contraction formula

The length contraction formula is given by: $$ L = L_0\sqrt{1 - \frac{v^2}{c^2}} $$ where \(L\) is the contracted length, \(L_0\) is the original length (in this case, 1 meter or 100 cm), \(v\) is the relative speed of the observer to the meter stick, and \(c\) is the speed of light.

Step 2: Plug in the given values

We want the length to be \(90.0 \mathrm{~cm}\), so substitute it into the equation: $$ 90 = 100\sqrt{1 - \frac{v^2}{c^2}} $$

Step 3: Solve for \(v^2\)

Divide both sides of the equation by 100, and then square both sides to get rid of the square root: $$ \frac{9}{10} = \sqrt{1 - \frac{v^2}{c^2}} $$ $$ \left(\frac{9}{10}\right)^2 = 1 - \frac{v^2}{c^2} $$

Step 4: Solve for \(v\)

Rearrange the equation to have \(v^2\) terms on the left and the constants on the right, then take the square root: $$ v^2 = c^2 \left(1 - \left(\frac{9}{10}\right)^2\right) $$ $$ v = c \sqrt{1 - \left(\frac{9}{10}\right)^2} $$

Step 5: Calculate the speed

Now, substitute the speed of light, \(c = 3 \times 10^8 \mathrm{\frac{m}{s}}\), into the equation and compute the value of \(v\): $$ v = (3 \times 10^8 \mathrm{\frac{m}{s}}) \sqrt{1 - \left(\frac{9}{10}\right)^2} $$ $$ v \approx 1.35 \times 10^8 \mathrm{\frac{m}{s}} $$ Therefore, the length of the meter stick will look \(90.0 \mathrm{~cm}\) when it is moving at a speed of approximately \(1.35 \times 10^8 \mathrm{\frac{m}{s}}\).

Key Concepts

Special Relativity

Special relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. It describes the physics of objects moving at significant fractions of the speed of light, especially how these objects behave differently from what we observe at everyday speeds. This theory changed our understanding of time, space, and motion.

In special relativity, time and space are not fixed but are relative to the observer's state of motion. One of the key concepts is that the laws of physics are the same for all non-accelerating observers, and the speed of light is constant in all inertial frames of reference. This means that no matter how fast you are moving, you will always measure the speed of light to be the same.

• Objects appear to contract in the direction of motion when they move close to the speed of light, a phenomenon known as length contraction.
• Time appears to pass slower for a moving observer from the perspective of a stationary observer, known as time dilation.

Special relativity becomes important when dealing with speeds that are a significant portion of the speed of light, like in high-energy physics experiments and astrophysics.

Speed of Light

The speed of light, denoted as \(c\), is a universal constant in physics. It is the fastest speed at which information can travel through the universe, acting as a cosmic speed limit. The value of the speed of light in a vacuum is approximately \(3 \times 10^8 \text{ m/s}\). This speed remains constant regardless of the motion of the source or the observer.

The constancy of the speed of light is a cornerstone of special relativity. It means that no matter how fast you're traveling, you will always observe light moving at \(c\). This unique characteristic leads to intriguing consequences like length contraction and time dilation.

• In modern physics, the speed of light ties together space and time into a single entity known as "spacetime," where the geometry of spacetime depends on the relative motion of both observers and objects.
• The constant speed of light enables us to define units of length and time precisely and consistently across different frames of reference.

Understanding the speed of light also aids in our comprehension of phenomena such as light's behavior in different media, how it bends in gravitational fields, and its role in the theory of general relativity.

Relativistic Physics

Relativistic physics refers to the study of phenomena that occur when an object approaches the speed of light. It extends classical mechanics by incorporating the effects predicted by Einstein's theory of relativity.

In relativistic physics, classic concepts like mass and energy are intertwined differently. Mass increases with speed, becoming infinite as an object approaches the speed of light. This means that achieving or surpassing the speed of light would require infinite energy, thus making it impossible for any material object.

Relativistic effects significantly impact calculations in physics, such as:

• The relativistic version of momentum and energy equations, which account for the increase in mass and energy as objects move faster.
• Adjustments in predictions for outcomes in particle physics experiments, where particles are often accelerated to near-light speeds.
• Correcting measurements for astronomical phenomena, allowing for more accurate models of the universe's dynamics.

Understanding these principles of relativistic physics ensures precise applications across advanced scientific fields, from designing technology for space travel to interpreting the behavior of cosmic phenomena.