Question

Robert, standing at the rear end of a railroad car of length \(100 . \mathrm{m},\) shoots an arrow toward the front end of the car. He measures the velocity of the arrow as \(0.300 c\). Jenny, who was standing on the platform, saw all of this as the train passed her with a velocity of \(0.750 c .\) Determine the following as observed by Jenny: a) the length of the car b) the velocity of the arrow c) the time taken by arrow to cover the length of the car d) the distance covered by the arrow

Short answer

Question: In an experiment by Robert and Jenny, a train car with a proper length of 100 meters is moving at a speed of 0.750c relative to Jenny. Inside the train, Robert shoots an arrow in the same direction as the train at a velocity of 0.300c relative to the train. Calculate the observed length of the car, the velocity of the arrow, the time taken by the arrow to cover the length of the car, and the distance covered by the arrow at Jenny's perspective. Answer: Based on the step by step solution provided, we have the following results: 1. The observed length of the car (L) by Jenny is approximately 66.67 meters. 2. The velocity of the arrow (u) as observed by Jenny is approximately 0.840c (84% of the speed of light). 3. The time taken by the arrow to cover the length of the car (t) is approximately 79.43 nanoseconds. 4. The distance (D) covered by the arrow according to Jenny is approximately 66.67 meters.

Step-by-step solution

1. Finding the length of the car observed by Jenny

Using the Lorentz contraction formula, we can find the length of the car (L) observed by Jenny. The equation for length contraction is: $$L = L_0 \sqrt{1 - (\frac{v}{c})^2}$$ where \(L_0\) is the proper length (100 m), \(v\) is the velocity of the train (0.750c), and \(c\) is the speed of light. Plug in the values and solve for L.

2. Finding the velocity of the arrow observed by Jenny

To find the velocity of the arrow observed by Jenny, we need to use the relativistic velocity addition formula: $$u = \frac{u' + v}{1 + \frac{u'v}{c^2}}$$ where \(u'\) is the arrow's velocity measured by Robert (0.300c), \(v\) is the velocity of the train (0.750c), and \(c\) is the speed of light. Plug in the values and solve for u - the velocity of the arrow as observed by Jenny.

3. Finding the time taken by the arrow to cover the length of the car

During the time taken by the arrow to cover the length of the car, Jenny observes the arrow and the car to be moving. The formula for calculating this time is a combination of distance, velocity, and time (t): $$L = u \cdot t$$ We have the observed length of the car (L) and the observed velocity of the arrow (u). Solve for t, the time taken by the arrow to cover the length of the car.

4. Finding the distance covered by the arrow

Lastly, to find the distance (D) covered by the arrow according to Jenny, we need to consider the relative velocities of the arrow and the train. Use the same formula as in step 3, but replace L with D and use the proper velocity of the arrow (u): $$D = u \cdot t$$ We know the time taken by the arrow (t) and the observed velocity of the arrow (u). Solve for D, the distance covered by the arrow.

Key concepts

Lorentz Contraction

Understanding Lorentz contraction is crucial when studying the behavior of objects moving at speeds close to that of light. According to the theory of relativity, objects appear to shorten in the direction of their motion when observed from a stationary frame of reference. For Jenny, standing still while the train passes at high-speed, this would make the train—and anything on it—seem shorter than it would appear to someone on the train, like Robert.
The Lorentz contraction formula is given by
\[ L = L_0 \sqrt{1 - \left(\frac{v}{c}\right)^2} \]
where \( L_0 \) is the object's rest length, \( v \) is the velocity of the moving object relative to the observer, and \( c \) is the speed of light. As the speed of the moving object approaches the speed of light, the contraction becomes more significant. It's a fascinating glimpse into how our classical notions of space are challenged at relativistic speeds.

Speed of Light

The speed of light in a vacuum, denoted by \( c \), is a fundamental constant of nature that is roughly \( 299,792,458 \) meters per second. It is the speed at which all massless particles and associated fields—including electromagnetic waves such as light—travel in a vacuum. This speed sets the scale for the universe's speed limit, meaning nothing with mass can travel faster than light. In the context of the exercise, both Robert's measurement of the arrow's speed and Jenny's observation of the train speed involve fractions of the speed of light, indicating the necessity to consider relativistic mechanics to understand the situation properly.

Relativistic Effects

When objects move at speeds comparable to the speed of light, Newtonian mechanics no longer provide accurate predictions. Instead, relativistic mechanics take the stage, describing phenomena like time dilation, length contraction, and the relativity of simultaneity. These relativistic effects enrich our understanding of time and space at high velocities.

In Jenny's observation of the train, relativistic effects play a significant role in determining the length of the train (Lorentz contraction), the speed of the arrow as it travels forward (relativistic velocity addition), and even the time measured for events occurring on the train (time dilation).

Time Dilation

Time dilation is another stunning prediction of Einstein’s theory of relativity. It states that time as measured by a clock moving relative to an observer will tick slower than a clock at rest with respect to that observer. This effect becomes more pronounced as the relative speed approaches the speed of light. For Jenny watching the train, a clock moving with the train, if observed, would appear to run slower compared to her own stationary clock.
Within the exercise scenario, we use the relationship \( L = u \cdot t \) to describe the time taken by the arrow to cover the perceived contracted length of the train. Time dilation would suggest that if Robert were to time the arrow's flight with a stopwatch, Jenny would observe this stopwatch to be running slow, thus taking more time for the same event.