Problem 70 - Fundamentals of Physics

Question

An airplane has rest length \(40.0 \mathrm{~m}\) and speed \(630 \mathrm{~m} / \mathrm{s} .\) To a ground observer, (a) by what fraction is its length contracted and (b) how long is needed for its clocks to be \(1.00 \mu \mathrm{s}\) slow?

Step-by-Step Solution

Answer

a) Length contraction is approximately \( 1 \times 10^{-6} \). b) Time required is \( 1 \, s \).

Step 1: Determine Lorentz Factor

To calculate the contraction in length and time dilation, we need the Lorentz factor \( \gamma \), given by the equation: \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \( v = 630 \mathrm{~m/s} \) and \( c = 3.00 \times 10^8 \mathrm{~m/s} \).Substituting the values:\[ \gamma = \frac{1}{\sqrt{1 - \frac{(630)^2}{(3.00 \times 10^8)^2}}} \approx 1.000001 \]

Step 2: Calculate Length Contraction

The fraction by which the length is contracted is given by \( 1 - \frac{1}{\gamma} \).Substitute the \( \gamma \) value:\[ 1 - \frac{1}{1.000001} \approx 1 \times 10^{-6} \]

Step 3: Calculate Time Dilation

For time dilation, we use the relation:\[ \Delta t = \frac{t_0}{\gamma} \]Given that the time difference \( \Delta t \) is \( 1.00 \mu s \), we rearrange:\[ t_0 = \Delta t \times \gamma = 1.00 \mu s \times 1.000001 \approx 1.00 \mu s \]

Step 4: Analyze Time Difference for Slowness

We need the onboard clock to slow down by \( 1.00 \mu s \). We need to calculate the time \( t \) it takes for the ground clock to register this:From the time dilation formula:\[ \Delta t = \gamma \Delta t_0 \]\[ \Delta t_0 = \frac{1.00 \mu s}{\gamma - 1} \approx 1.00 \times 10^{-6} / (1.000001 - 1) \approx 1.00 \times 10^{-6} / 1 \times 10^{-6} \approx 1000000 \mu s = 1 s \]

Step 5: Conclusion

a) The length is contracted by a fraction of approximately \( 1 \times 10^{-6} \) relative to the rest length.b) It takes approximately \( 1 \) second for the airplane’s clocks to be \( 1.00 \mu s \) slower than the ground clock due to time dilation.

Key Concepts

Lorentz Factor

The Lorentz factor, denoted as \( \gamma \), plays an essential role in special relativity. It quantifies how much relativistic effects like time dilation and length contraction will occur at a given velocity. The Lorentz factor is calculated using the formula: \[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\] where \( v \) is the velocity of the object and \( c \) is the speed of light. The closer the velocity \( v \) of an object is to the speed of light \( c \), the more pronounced the relativistic effects will be.

• At low velocities compared to the speed of light, \( \gamma \approx 1 \), indicating negligible relativistic effects.
• As velocity approaches the speed of light, \( \gamma \) increases significantly, leading to notable time dilation and length contraction.

In the airplane example, with a speed of 630 m/s, the calculated \( \gamma \approx 1.000001 \) suggests minimal relativistic effects due to its relatively low speed compared to \( c \). Thus, the Lorentz factor helps in adjusting the perception of time and space in moving objects.

Length Contraction

Length contraction is a phenomenon in special relativity that describes how an object's length appears shorter when it moves at high speeds, especially those comparable to the speed of light. The contraction only occurs in the direction of the movement.
Using the Lorentz factor, length contraction can be calculated with the formula: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where \( L \) is the contracted length observed, \( L_0 \) is the rest length, and \( v \) and \( c \) are the object's velocity and the speed of light, respectively. Alternatively, the fractional contraction can be expressed as \( 1 - \frac{1}{\gamma} \).

• At low speeds (like the airplane at 630 m/s), the contraction is almost negligible.
• The fraction of contraction is approximately \( 1 \times 10^{-6} \), meaning the rest length and contracted length are nearly identical from a practical perspective.

This demonstrates that while special relativity predicts length contraction, its effects are significant primarily at relativistic speeds.

Time Dilation

Time dilation is another intriguing effect of special relativity, where time seems to pass slower for objects moving at high speeds compared to stationary observers. The faster the object moves, the greater the time dilation observed. This effect becomes relevant as an object approaches the speed of light.
Time dilation is calculated using the formula: \[ \Delta t = \gamma \cdot \Delta t_0 \] where \( \Delta t \) is the dilated time observed by a stationary observer, \( \Delta t_0 \) is the proper time measured by a moving observer, and \( \gamma \) is the Lorentz factor.
For the airplane, although its speed is much lower than the speed of light, we can still calculate time dilation. The onboard clocks are slowed by \( 1.00 \mu s \) according to ground time.

• Using \( \gamma \approx 1.000001 \), it takes around 1 second on the ground for the airplane's clocks to lag by \( 1.00 \mu s \).
• This small amount of dilation shows how subtle the effect is at lower speeds.

Time dilation explains why travelers moving close to light speed could experience less passage of time compared to those standing still.