Relativistic dynamics

In summary: The increase in energy is the work done, so you just use the difference in kinetic energy.In summary, the question asks for the amount of work required to increase the speed of a proton from given fractions of the speed of light (0.15c, 0.81c, and 0.93c) to slightly higher fractions (0.16c, 0.82c, and 0.94c). The equation used is E = 1/2 m v^2, with a small correction for relativistic speeds. The work done is equal to the increase in kinetic energy, which can be converted from Joules to MeV for easier understanding.
  • #1
patapat
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Homework Statement


How much work must be done on a proton to increase its speed by each of the values below?

(a) 0.15c to 0.16c
(b) 0.81c to 0.82c
(c) 0.93c to 0.94c

The answer is given in MeV which is a million electron volts.
mass of proton=1.67x10^-27kg

Homework Equations


e=mc^2
1 J=1.6x10^-19eV=1.6x10^-13MeV

The Attempt at a Solution


I'm not sure if this is even the correct equation to use, but I don't see any other appropriate equations in this book.

Energy required at .15c
e=(1.67x10^-27kg)(.15*3x10^8)^2
e=3.38x10^-12J
Convert to MeV
e=21.14MeV

Energy required at .16c
e=(1.67x10^-27kg)(.16*3x10^8)^2
e=3.85x10^-12J
convert to MeV
e=24.05MeV

Energy required to go from .15c-16c
24.05MeV-21.14MeV=2.91MeV
Which I found to be wrong, obviously.

Thanks in advance to any and all help. Much appreciated.

-Pat
 
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  • #2
You have misunderstood the equation, in e=mc^2 C is the speed of light and is always constant. The value s0.15C etc that you have been given are speeds as fractions of the speed of light.
Kinetic energy is E = 1/2 m v^2, there is a small correction for relativistic speeds which you will have to lookup for the higher spped cases.
 
  • #3
rick

Dear Patrick,

Thank you for your question. It seems that you are on the right track with your calculations, but there are a few things that need to be corrected.

First, the equation e=mc^2 is indeed the correct equation to use for this problem. However, you need to use the relativistic mass of the proton, which takes into account its increase in mass as it approaches the speed of light. The equation for relativistic mass is m = m0/(1-v^2/c^2)^1/2, where m0 is the rest mass of the proton, v is its velocity, and c is the speed of light. So for the first part of the problem, the mass of the proton at 0.15c would be:

m = (1.67x10^-27kg)/(1-(0.15*3x10^8)^2/(3x10^8)^2)^1/2
m = 1.71x10^-27kg

Next, you need to use the equation for kinetic energy, which is K = (m - m0)c^2. So the energy required to increase the speed from 0.15c to 0.16c would be:

K = (1.71x10^-27kg - 1.67x10^-27kg)(3x10^8)^2
K = 1.44x10^-13J

To convert this to MeV, you need to divide by 1.6x10^-13 MeV/J to get 0.9 MeV. So the energy required to increase the speed from 0.15c to 0.16c is approximately 0.9 MeV.

Using this same method, you can calculate the energy required for the other two parts of the problem. For part (b), the energy required would be approximately 5.9 MeV, and for part (c), it would be approximately 30 MeV.

I hope this helps clarify the problem for you. Please let me know if you have any further questions.

Best,
 

1. What is the theory of Relativistic Dynamics?

The theory of Relativistic Dynamics is a branch of physics that studies the motion of objects at high speeds and in the presence of strong gravitational fields. It combines the principles of Special Relativity and classical mechanics to describe the behavior of objects in these extreme conditions.

2. What is Special Relativity and how does it relate to Relativistic Dynamics?

Special Relativity is a theory proposed by Albert Einstein that explains the relationship between space and time in the absence of gravitational forces. It is the foundation of Relativistic Dynamics, as it provides the framework for understanding the behavior of objects at high speeds and in different frames of reference.

3. How does Relativistic Dynamics differ from classical mechanics?

Relativistic Dynamics takes into account the principles of Special Relativity, such as time dilation and length contraction, which are not included in classical mechanics. It also considers the effects of gravity and the curvature of spacetime, which are not significant in classical mechanics.

4. What are some practical applications of Relativistic Dynamics?

Relativistic Dynamics has been used to explain and predict phenomena such as the behavior of particles in particle accelerators, the motion of objects in space, and the behavior of black holes. It also has practical applications in technologies such as GPS systems and nuclear reactors.

5. Can you provide a real-life example of Relativistic Dynamics in action?

One famous example of Relativistic Dynamics is the twin paradox, where one twin travels at high speeds in space while the other stays on Earth. When the traveling twin returns, they have aged less than their twin on Earth due to the effects of time dilation. This phenomenon has been observed and confirmed through experiments with atomic clocks on airplanes.

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