Question

How fast must a rocket travel relative to the earth so that time in the rocket "slows down" to half its rate as measured by earthbased observers? Do present- day jet planes approach such speeds?

Short answer

The rocket must travel at 0.866 times the speed of light; present-day jets do not reach such speeds.

Step-by-step solution

Understanding Time Dilation

The phenomenon in question is known as time dilation, which is described by Einstein's theory of relativity. According to this theory, time moves slower for an object that is moving relative to a stationary observer. The formula used to calculate time dilation is:\[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \( t' \) is the time measured in the moving frame (rocket), \( t \) is the time measured in the stationary frame (earth), \( v \) is the velocity of the moving frame relative to the stationary observer, and \( c \) is the speed of light.

Setting Up the Equation

In this problem, we are asked to find the speed \( v \) at which time in the rocket slows down to half the rate as measured by earth-based observers. This means \( t' = 0.5t \). Substituting this into the time dilation formula, we can write:\[ 0.5t = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Solving for Velocity

Simplify and solve for \( v \):1. Cancel \( t \) from both sides:\[ 0.5 = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]2. Invert both sides:\[ \sqrt{1 - \frac{v^2}{c^2}} = 2 \]3. Square both sides to eliminate the square root:\[ 1 - \frac{v^2}{c^2} = \frac{1}{4} \]4. Rearrange the equation to solve for \( \frac{v^2}{c^2} \):\[ \frac{v^2}{c^2} = 1 - \frac{1}{4} = \frac{3}{4} \]5. Solve for \( v \):\[ v = c \sqrt{\frac{3}{4}} = c \frac{\sqrt{3}}{2} \]

Conclusion and Comparison

The derived formula indicates that for the time in the rocket to slow down to half its rate as measured by observers on Earth, the rocket must travel at a speed of \( c \frac{\sqrt{3}}{2} \), which is approximately 0.866 times the speed of light. Present-day jet planes travel at a maximum speed of about 1000 km/h, which is extremely far from approaching the speed of light at roughly 1,080,000,000 km/h.

Key concepts

Einstein's theory of relativity

Einstein's theory of relativity revolutionized our understanding of time and space. It brought forth the idea that the laws of physics are the same for all non-accelerating observers and revealed that the speed of light in a vacuum is constant, regardless of the motion of the light source or observer. This includes both special relativity, dealing with objects moving at constant speeds, and general relativity, which incorporates gravity.
This theory upended previous notions of absolute time and space, replacing them with the concept of spacetime. In spacetime, the fabric of reality is affected by the presence of mass and energy. Therefore, both time and space are relative, meaning they can change based on how fast an object is moving and how much gravitational force is being exerted on it.
Understanding this theory is central to grasping the concept of time dilation, a cornerstone phenomenon that occurs at high velocities or in strong gravitational fields. Time dilation reveals that a clock moving relative to an observer will tick slower than a clock at rest with respect to that observer. This fascinating outcome is a direct consequence of relativity, hinting at how intertwined our perceptions of time and space truly are.

speed of light

The speed of light is a fundamental constant in modern physics, denoted by the letter "c." It is approximately equal to 299,792,458 meters per second (m/s) or about 186,282 miles per second. The invariance of this speed is a critical element in Einstein's special theory of relativity.
The speed of light is the ultimate speed limit of the universe. Nothing with mass can reach or exceed this speed due to the infinite energy required to do so. At speeds nearing that of light, an object's mass effectively becomes infinitely large, demanding more and more energy to continue accelerating.
This constant plays a key role in equations that describe time dilation, linking time, velocity, and distance in the universe. Consequently, the speed of light not only limits how fast we can send information or travel through space but also shapes the cosmic scale of events and our perception of the universe.

velocity calculations

Velocity calculations are essential for understanding relativistic effects like time dilation. In the formula for time dilation, velocity ( v ) shows how an object's speed relative to an observer affects the rate at which time passes in different reference frames. The equation: \[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \] allows us to determine the degree of time dilation due to an object's speed.

• As velocity approaches the speed of light, the denominator in the time dilation equation decreases, leading to greater time dilation effects (time appears to pass slower).
• For velocity calculations in these high-speed scenarios, it's crucial to express the velocity as a fraction of the speed of light to correctly apply relativistic physics.

To calculate the velocity necessary for specific time dilation effects, such as slowing time by half, requires manipulating the equation to solve for v . By doing so, one can determine at what fraction of light speed a certain relativistic effect occurs, highlighting just how fast that is compared to everyday speeds.

modern physics education

Modern physics education encompasses understanding complex theories and concepts such as time dilation, which require a strong foundation in mathematics and scientific principles. Educators strive to bring Einstein's revolutionary ideas to life through interactive learning experiences.
To grasp these advanced topics, students typically engage with practical exercises and visualizations that illustrate abstract concepts. For instance, simulations can demonstrate how spacetime is affected by massive objects and speeds approaching that of light. Learning platforms often employ digital tools to break down these theories into digestible pieces.
Additionally, discussions around the implications of these scientific ideas in technology, such as GPS systems which account for time dilation due to their high speeds and altitudes, make modern physics education relevant to everyday life. By connecting theoretical concepts to real world applications, students gain a deeper appreciation for the impact of modern physics on contemporary technology and worldviews.