# Special relativity - velocity additions/energy

In summary: You need to use the relativistic kinetic energy equation, E = mc^2 + K. Use the Lorentz transformation to find the velocity and then plug it into the equation.

## Homework Statement

1. A spaceship traveling away from Earth with a speed of 0.6c as measured by an observer on earth. The rocket sends back a light pulse back to Earth every 10 minutes as measured by a clock in the spaceship.
How far does the spaceship travel between each light pulses as measured by: a) the observer on Earth and b) someone on the spaceship?

2. An electron is moving at a constant velocity of 0.90c with respect to a lab observer X. Another observer Y is moving at a constant velocity of 0.50c with respect to X in a direction opposite to that of the electron in X's reference frame. What is the mass of the electron as measured be Y?

3. Protons are accelerated from rest through a potential difference of 8.0E8 V. Calculate as measured in the laboratory frame of reference after acceleration the proton's mass, velocity, momentum and total energy.

## Homework Equations

Lorentz transformation?

## The Attempt at a Solution

1. a) I first found time t passed on Earth between pulses and simply used d = (t)(0.6c). But it didn't work. And I haven't a clue how to do the second part.

The other ones I don't have a single clue.

These equations might help:

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

$$w'=\frac{w-v}{1-wv/c^2}$$

If the observer in S sees an object moving along the x-axis at velocity w, then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity w'.

$$M = \frac{E}{c^2}\!$$

$$m_{\mathrm{rel}} = { m \over \sqrt{1-{v^2\over c^2}}}$$

$m_{\mathrm{rel}}$=relativistic mass.

Oh and for part one you might want to convert .6c into m/s so you are using the same units. Might make it easier.

Edit:

Oh and:

$$\ t' = t-vx/c^2$$

Last edited:

## The Attempt at a Solution

1. a) I first found time t passed on Earth between pulses and simply used d = (t)(0.6c). But it didn't work.
Why didn't it work? How did you find t?

Never mind, I figured it out.

But for the 2nd question, I'm confused. To find the mass, I first need to find the velocity of the electron relative to Y. How would I input the values into the equation? If Y is moving in opposite direction of the electron, then I should put its velocity as negative? And so U' = 0.9c - (-0.5c)/(1 - (-0.5c x 0.9c)/c^2)) = 1.4/1.45 = 0.97c
But the answer to the question says that the relative velocity is 0.73c...

As for the 3rd question, I really don't even know how to approach this. I was thinking that E = Vq = mc^2 but my answer was totally off. The answer is 3.1 x 10^-27 kg for mass.

Last edited:
But for the 2nd question, I'm confused. To find the mass, I first need to find the velocity of the electron relative to Y. How would I input the values into the equation? If Y is moving in opposite direction of the electron, then I should put its velocity as negative? And so U' = 0.9c - (-0.5c)/(1 - (-0.5c x 0.9c)/c^2)) = 1.4/1.45 = 0.97c
But the answer to the question says that the relative velocity is 0.73c...
Your answer for the relative velocity is correct. An answer of 0.73c makes no sense, since you know the speed must be greater than 0.9c.

As for the 3rd question, I really don't even know how to approach this. I was thinking that E = Vq = mc^2 but my answer was totally off. The answer is 3.1 x 10^-27 kg for mass.
Vq represents an increase in the kinetic energy; it does not equal the total energy (or the rest energy).

## 1. What is special relativity and how does it relate to velocity additions?

Special relativity is a theory proposed by Albert Einstein in 1905 that describes how objects move at high speeds, close to the speed of light. It states that the laws of physics are the same for all observers in uniform motion. Velocity additions in special relativity refer to how velocities are combined when objects are moving close to the speed of light.

## 2. How does special relativity impact the concept of energy?

Special relativity has a significant impact on the concept of energy. It introduces the famous equation E=mc^2, which states that energy (E) is equal to mass (m) multiplied by the speed of light squared (c^2). This equation shows that mass and energy are interchangeable and that even a small amount of mass can contain a large amount of energy.

## 3. Can objects travel faster than the speed of light in special relativity?

No, according to special relativity, the speed of light is the fastest speed possible. As an object approaches the speed of light, its mass increases, making it more difficult to accelerate. Therefore, it is impossible for an object with mass to reach or exceed the speed of light.

## 4. How does special relativity affect time and distance measurements?

Special relativity introduces the concepts of time dilation and length contraction. Time dilation refers to how time appears to pass slower for objects moving at high speeds. Length contraction refers to how objects appear to become shorter in the direction of motion when moving at high speeds. These effects are only noticeable at speeds close to the speed of light.

## 5. What are some real-life applications of special relativity?

Special relativity has many practical applications, such as in the development of GPS technology. The satellites used in GPS rely on precise time measurements, which are affected by the time dilation effect of special relativity. Another application is in nuclear energy, where the conversion of mass to energy is used to produce electricity. Special relativity also plays a role in the study of particle physics and the behavior of particles at high energies.

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