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Chapter 15: Problem 24

A robber's getaway vehicle, which can travel at an impressive 0.8c, is pursued by a cop, whose vehicle can travel at a mere 0.4c. Realizing that he cannot catch up with the robber, the cop tries to shoot him with bullets that travel at \(0.5 c\) (relative to the cop). Can the cop's bullets hit the robber?

Short Answer

Expert verified
No, the bullets cannot reach the robber; their relative speed is negative.

Step by step solution

01

Understand the Scenario

In this physics problem, we have three main objects: the robber's car moving at 0.8c, the cop's car moving at 0.4c, and bullets from the cop's gun traveling at 0.5c relative to the cop. We need to determine if these bullets can reach the robber's car.
02

Apply the Relativistic Velocity Addition Formula

To find the speed of the bullet relative to the robber, we need to use the relativistic velocity addition formula: \( u' = \frac{u + v}{1 + \frac{uv}{c^2}} \) where \( u \) is the bullet's speed relative to the cop, \( v \) is the speed of the cop relative to the robber, and \( c \) is the speed of light.
03

Determine the Relative Velocity of the Bullet

Let \( u = 0.5c \) (bullet speed relative to cop) and \( v = -0.4c \) (cop's speed relative to the robber, negative because they are moving in opposite directions). Substitute these into the formula: \[ u' = \frac{0.5c - 0.8c}{1 - \frac{(0.5c)(0.8c)}{c^2}}\]
04

Calculate the Result

Simplify the expression:Combine the numerators: \(0.5c - 0.8c = -0.3c\).Calculate the denominator: \(1 - \frac{(0.5 \times 0.8)c^2}{c^2} = 1 - 0.4 = 0.6\).Now compute \( u' \):\[ u' = \frac{-0.3c}{0.6} = -0.5c\]This means the bullet travels at a speed of \(-0.5c\) relative to the robber.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz Transformations
Lorentz transformations play a crucial role when discussing movements at relativistic speeds, especially as speeds approach the speed of light, denoted as \(c\). They help us convert the coordinates and times in one reference frame to another moving at constant velocity relative to the first. These transformations highlight how time and space are interconnected and can appear different from different observational frames.

In our scenario, the robber and the cop are moving at significant fractions of the speed of light, making relativistic equations necessary. At these speeds, classical physics cannot accurately describe the situation. Lorentz transformations allow us to understand how fast objects move relative to each other, considering time dilation and length contraction, which are key elements in relativistic physics.

Using the relativistic velocity addition formula is an application of Lorentz transformations in determining how two velocities combine when both are significant fractions of the speed of light. This is essential here to find the bullet's speed relative to the robber, as simple addition or subtraction of velocities would not give the correct result.
Special Relativity
Special relativity, formulated by Albert Einstein, revolutionized how we understand motion and the structure of space and time. It is based on two main principles: the laws of physics are the same in all inertial frames of reference, and the speed of light in a vacuum is constant and not dependent on the motion of the light source or observer.

In the given problem, special relativity provides the tools to address the question of whether the cop's bullets can catch up with the robber's vehicle. According to special relativity, as objects move closer to the speed of light, their mass effectively increases, and their behavior must be described by relativistic effects rather than classical mechanics.

The physics of the situation doesn鈥檛 follow intuitive, everyday rules, but rather requires careful calculations as demonstrated in the step-by-step solution. Understanding these concepts ensures we correctly predict phenomena occurring at high velocities.
Relativistic Effects
Relativistic effects profoundly influence how objects behave at speeds approaching the speed of light. These effects are non-intuitive because they disregard our everyday experiences of motion. A key relativistic effect is the inability to simply add speeds together. Instead, we use the relativistic velocity addition formula to account for the changes in time and space as perceived by different observers.

In this problem, the main relativistic effect is evident in how we calculate the speed of the bullets relative to the robber. Despite the bullets traveling at 0.5c relative to the cop, the resulting speed relative to the robber, calculated through the relativistic velocity addition formula, is -0.5c.

This result indicates that the bullets are actually moving away from the robber, rather than towards him. This counterintuitive outcome showcases the need for relativistic physics when dealing with high-speed pursuits like this one. In summary, relativistic effects highlight the complexities of high-velocity interactions and serve as a reminder that space and time are intertwined dimensions.

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Most popular questions from this chapter

Consider two events that occur simultaneously at \(t=0\) in frame \(\mathcal{S},\) both on the \(x\) axis at \(x=0\) and \(x=a\). (a) Find the times of the two events as measured in a frame \(\mathcal{S}^{\prime}\) traveling in the positive direction along the \(x\) axis with speed \(V\). (b) Do the same for a second frame \(\mathcal{S}^{\prime \prime}\) traveling at speed \(V\) but in the negative direction along the \(x\) axis. Comment on the time ordering of the two events as seen in the three different frames. This startling result is discussed further in Section 15.10 .

A rocket traveling at speed \(\frac{1}{2} c\) relative to frame \(\mathcal{S}\) shoots forward bullets traveling at speed \(\frac{3}{4} c\) relative to the rocket. What is the speed of the bullets relative to \(\mathcal{S} ?\)

Like time dilation, length contraction cannot be seen directly by a single observer. To explain this claim, imagine a rod of proper length \(l_{\mathrm{o}}\) moving along the \(x\) axis of frame \(\mathcal{S}\) and an observer standing away from the \(x\) axis and to the right of the whole rod. Careful measurements of the rod's length at any one instant in frame \(\mathcal{S}\) would, of course, give the result \(l=l_{\mathrm{o}} / \gamma\). (a) Explain clearly why the light which reaches the observer's eye at any one time must have left the two ends \(A\) and \(B\) of the rod at different times. (b) Show that the observer would see (and a camera would record) a length more than \(l\). [It helps to imagine that the \(x\) axis is marked with a graduated scale.] ( \(\mathbf{c}\) ) Show that if the observer is standing close beside the track, he will see a length that is actually more than \(l_{\mathrm{o}}\); that is, the length contraction is distorted into an expansion.

A positive pion decays at rest into a muon and neutrino, \(\pi^{+} \rightarrow \mu^{+}+\nu .\) The masses involved are \(m_{\pi}=140 \mathrm{MeV} / c^{2}, m_{\mu}=106 \mathrm{MeV} / c^{2},\) and \(m_{\nu}=0 .\) (There is now convincing evidence that \(m_{\nu}\) is not exactly zero, but it is small enough that you can take it to be zero for this problem.) Show that the speed of the outgoing muon has \(\beta=\left(m_{\pi}^{2}-m_{\mu}^{2}\right) /\left(m_{\pi}^{2}+m_{\mu}^{2}\right) .\) Evaluate this numerically. Do the same for the much rarer decay mode \(\pi^{+} \rightarrow \mathrm{e}^{+}+\nu,\left(m_{\mathrm{e}}=0.5 \mathrm{MeV} / c^{2}\right)\)

Show that any two zero-mass particles have a CM frame, provided their three- momenta are not parallel. [Hint: As you should explain, this is equivalent to showing that the sum of two forward light-like vectors is forward time-like, unless the spatial parts are parallel.]
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