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Chapter 38: Problem 29

(a) Find an equation for the unknown mass \(m\) of a particle if you know its momentum \(p\) and its kinetic energy \(K\). Show that this expression reduces to an expected result for nonrelativistic particle speeds. (b) Find the mass of a particle whose kinetic energy is \(K=55.0 \mathrm{MeV}\) and whose momentum is \(p=\) \(121 \mathrm{MeV} / \mathrm{c}\). Express your answer as a decimal fraction or multiple of the mass \(m_{\mathrm{e}}\) of the electron.

Short Answer

Expert verified
The mass is approximately 206 times the electron mass.

Step by step solution

01

- Write down the given quantities

We know the momentum of the particle is given: \[ p \]and its kinetic energy is:\[ K \].
02

- Use the relationship between energy and mass

We use the relativistic relationship:\[ E^2 = (pc)^2 + (mc^2)^2 \]where \[E = K + mc^2 \].
03

- Substitute kinetic energy expression into the energy equation

Substitute \[ E = K + mc^2 \] into \[ E^2 = (pc)^2 + (mc^2)^2 \] to get:\[ (K + mc^2)^2 = (pc)^2 + (mc^2)^2 \].
04

- Solve for mass \(m\)

Expand the equation and solve for \[m\]:\[ K^2 + 2Kmc^2 + m^2c^4 = p^2c^2 + m^2c^4 \]which simplifies to:\[ K^2 + 2Kmc^2 = p^2c^2 \]then:\[ m = \frac{p^2c^2 - K^2}{2Kc^2} \].
05

- Reduce to nonrelativistic limit

For nonrelativistic speeds, where \[ K \approx \frac{p^2}{2m} \]. substituting \[ K \] into the mass equation:\[ K \approx \frac{p^2}{2m} \Rightarrow m \approx \frac{p^2}{2Kc^2} \].
06

- Calculate mass using given values

Using the given values \[ K = 55.0 \, \mathrm{MeV} \] and \[ p = 121 \, \mathrm{MeV}/c \], substitute into the formula:\[ m = \frac{(121 \, \mathrm{MeV}/c)^2 - (55.0 \, \mathrm{MeV})^2}{2 \times 55.0 \, \mathrm{MeV} \times c^2} \].
07

- Calculate the numerical result

Compute the values:\[ m = \frac{14641 - 3025}{110c^2} \, \mathrm{MeV}/c^2 \]which simplifies to:\[ m \approx 105.1 \mathrm{MeV}/c^2 \].
08

- Express the mass as a multiple of the electron mass

Given mass of electron \[m_e = 0.511 \mathrm{MeV}/c^2\]:\[ m = \frac{105.1 \mathrm{MeV}/c^2}{0.511 \mathrm{MeV}/c^2} \approx 206 \].

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Key Concepts

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

Relativity
Relativity is a fundamental concept in physics that revolutionized our understanding of space, time, and energy. It's described by Einstein's theory of Special Relativity, which explains how objects behave at high speeds close to the speed of light. One crucial aspect to understand is that as an object's speed increases, its mass appears to increase from the perspective of an observer. This is termed relativistic mass. The famous equation from this theory, \(E = mc^2\), shows that mass and energy are interchangeable. In our exercise, we use the relativistic energy equation \[ E^2 = (pc)^2 + (mc^2)^2 \] to find the mass of a particle given its momentum and kinetic energy. This equation helps tie together energy (E), momentum (p), and mass (m), showcasing the beauty of relativity.
Kinetic energy
Kinetic energy is the energy of motion. An object possesses kinetic energy when it is moving. For particles moving at or near the speed of light, their kinetic energy is calculated using the relativistic energy equation. This takes into account not only their speed but also their mass and the speed of light. The kinetic energy (K) in relativity is given by \[ K = E - mc^2 \]. In the exercise, we worked with a particle with known kinetic energy and momentum. Using this information, we could derive the unknown mass by incorporating kinetic energy into the relativistic energy equation. For very high speeds, kinetic energy can significantly impact the calculated mass.
Momentum
Momentum is the product of an object's mass and velocity. It describes how much motion an object has and how much force is needed to change that motion. In relativistic physics, momentum becomes more complex because it increases with velocity near the speed of light. The relativistic momentum is given by \( p = \frac{mv}{\beta} \) where \( \beta = \frac{v}{c} \) and \( c \) is the speed of light. In the given exercise, we use the momentum (p) of the particle to aid in calculating its mass. By combining knowledge of momentum and kinetic energy, we apply the relativistic formulas to find a consistent solution.
Energy equation
The energy equation is a pivotal component when dealing with relativistic particles. It interrelates momentum, kinetic energy, and mass. Specifically, we use the equation \[ E^2 = (pc)^2 + (mc^2)^2 \]. This formula incorporates the critical aspects of relativity by accounting for the total energy (E) of a particle as a combination of its momentum energy \( (pc)^2 \) and its rest mass energy \( (mc^2)^2 \). In solving the exercise, we substitute the kinetic energy expression \( E = K + mc^2 \) into this equation. By isolating mass (m), we can calculate the exact mass of a particle given specific values for kinetic energy and momentum.

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Most popular questions from this chapter

You see a sudden eruption on the surface of the Sun. From solar theory you predict that the eruption emitted a pulse of particles that is moving toward the Earth at oneeighth the speed of light. How long do you have to seek shelter from the radiation that will be emitted when the particle pulse hits the Earth? Take the light-travel time from the Sun to the Earth to be 8 minutes.

An aspirin tablet contains 5 grains of aspirin (medicinal unit), which is equal to \(325 \mathrm{mg}\). For how many kilometers would the energy equivalent of this mass power an automobile? Assume \(12.75 \mathrm{~km} / \mathrm{L}\) and a heat of combustion of \(3.65 \times 10^{7} \mathrm{~J} / \mathrm{L}\) for the gasoline used in the automobile.

Quite apart from effects due to the Earth's rotational and orbital motion, a laboratory reference frame on the Earth is not an inertial frame, as required by a strict interpretation of special relativity. It is not inertial because a particle released from rest at the Earth's surface does not remain at rest; it falls! Often, however, the events in an experiment for which one needs special relativity happen so quickly that we can ignore effects duc to gravitational accclcration. Considcr, for cxamplc, a proton moving horizontally at speed \(v=0.992 c\) through a 10 -m-wide detector in a laboratory test chamber. (a) How long will the transit through that detector take? (b) How far does the proton fall vertically during this time lapse? (c) What do you conclude about the suitability of the laboratory as an inertial frame in this case?

Evelvn Brown does not approve of our latticework of rods and clocks and the use of a light flash to synchronize them. (a) "I can synchronize my clocks in any way I choose!" she exclaims. Is she right? (b) Evelyn wants to synchronize two identical clocks, called Big Ben and Little Ben, which are at rest with respect to one another and separated by one million kilometers in their rest frame. She uses a third clock, identical in construction with the first two, that travels with constant velocity between them. As her moving clock passes Big Ben, it is set to read the same time as Big Ben. When the moving clock passes Little Ben, that outpost clock is set to read the same time as the traveling clock. "Now Big Ben and Little Ben are synchronized," says Evelyn Brown. Is Evelyn's method correct? (c) After Evelyn completes her synchronization of Little Ben by her method, how does the reading of Little Ben compare with the reading of a nearby clock on a latticework at rest with respect to Big Ben (and Little Ben) and synchronized by our standard method using a light flash? Evaluate in milliseconds any difference between the reading on Little Ben and the nearby lattice clock in the case that Evelyn's traveling clock moved at a constant velocity of 500000 kilometers per hour from Big Ben to Little Ben. (d) Evaluate the difference in the reading between the EvelynBrown- synchronized Little Ben and the nearby lattice clock when Evelyn's synchronizing traveling clock moves 1000 times as fast as the speed given in part (c).

A spaceship of rest length \(100 \mathrm{~m}\) passes a laboratory timing station in \(0.2\) microseconds measured on the timing station clock. (a) What is the speed of the spaceship in the laboratory frame? (b) What is the Lorentz-contracted length of the spaceship in the laboratory frame?
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