Relativistic calculation of speed when the momentum is known

In summary, the conversation discusses the relationship between momentum and speed of an electron, and how it can be calculated using the formula p = mv/√(1 - v^2/c^2). The steps to transform the formula and solve for v are explained, leading to the simplified equation v = (p*c) / √(p^2 + m^2*c^2).
  • #1
Karagoz
52
5

Homework Statement


We know the momentum of an electron, which is: 1,48*10^-21.

Momentum is m*v (mass*speed)

If we divide the momentum by the mass of the electron to find electron's speed, it'll give a value where v> 3*10^8 m/s.

Since speed can't be above speed of light, we have to calculate it relativistic to find the speed of light of the electron.

Homework Equations



upload_2018-5-2_17-39-35.png

gives:
upload_2018-5-2_17-39-50.png


The Attempt at a Solution



With the formula above, the problem is easy to solve.

But I don't get how that formula is transformed.

When I try it, this is what I get:

p = mv/√(1 - v^2/c^2)

divide by m, and multiply by √(1 - v^2/c^2)

v = p/m * √(1 - v^2/c^2)

^2 both sides

v^2 = p^2/m^2 * (1 - v^2/c^2) = p^2/m^2 - (p^2*v^2)/(c^2*m^2)

making some changes so both have same divsor:

v^2 = (p^2*c^2)/(m^2*c^2) - (p^2*v^2)/(m^2*c^2)

How do they get the equation where:

v = (p*c) / √(p^2 + m^2*c^2)

??
 

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  • #2
Karagoz said:
v^2 = (p^2*c^2)/(m^2*c^2) - (p^2*v^2)/(m^2*c^2)
Collect together the terms that contain v2 and factor out the v2.
(The toolbar has a superscript tool.)
 
  • #3
Start with your first Relevant equation and solve for v! Collect all the terms with v on one side, square everything to get rid of the square root, and solve for v2
 
  • #4
Karagoz said:
We know the momentum of an electron, which is: 1,48*10^-21.
Also, remember that numbers are useless without appropriate units.
 
  • #5
p = mv/√(1 - v^2/c^2)

p = mv/√(1 - v^2/c^2)

divide by m, and multiply by √(1 - v^2/c^2)

v = p/m * √(1 - v^2/c^2)

^2 both sides

v^2 = p^2/m^2 * (1 - v^2/c^2) = p^2/m^2 - (p^2*v^2)/(c^2*m^2)

making some changes on the right side so both have same divsor:

v^2 = (p^2*c^2)/(m^2*c^2) - (p^2*v^2)/(m^2*c^2)

This was where I left:

v^2 = (p^2*c^2)/(m^2*c^2) - (p^2*v^2)/(m^2*c^2)

Add + (p^2*v^2)/(m^2*c^2) on both sides.

v^2 + (p^2*v^2)/(m^2*c^2) = (p^2*c^2)/(m^2*c^2)

Get a common divisor on left side.

(v^2*m^2*c^2)/(m^2*c^2) + (p^2*v^2)/(m^2*c^2)

Simplify the left side

v^2(m^2*c^2 + p^2) / (m^2*c^2) = (p^2*c^2) / (m^2*c^2)

Multiply both sides by: (m^2*c^2)

v^2(m^2*c^2 + p^2) = (p^2*c^2)

divide by (m^2*c^2 + p^2)

v^2 = (p^2*c^2)/(m^2*c^2 + p^2)

SQROOT both sides:

v = (p*c) / √(m^2*c^2 + p^2)

v = (p*c) / √(p^2 + m^2*c^2)Do you know a more simplified way?
 
Last edited:
  • #6
p/m = v / √(1 - v2 / c2)
p2 / m2 = v2 / (1 - v2 / c2) = 1 / (1/v2 - 1/c2)
(1/v2 - 1/c2) = m2 / p2
1/v2 = (m2 / p2) + (1/c2))
That separates the v to the left. Now take the lcm on the right hand side, invert, and take the square root.
 

1. How is relativistic speed calculated when the momentum is known?

Relativistic speed can be calculated using the formula v = p/m, where v is the speed, p is the momentum, and m is the mass. This formula takes into account the effects of relativity, such as time dilation and length contraction.

2. What is the difference between classical and relativistic speed calculations?

In classical physics, speed is calculated using the formula v = d/t, where d is the distance traveled and t is the time taken. This does not take into account the effects of relativity. In contrast, relativistic speed calculations use the formula v = p/m, which takes into account the mass and momentum of an object.

3. What is the speed of light in a relativistic calculation?

In a relativistic calculation, the speed of light is always constant and is denoted by the symbol c. This value is approximately 299,792,458 meters per second in a vacuum and is considered the maximum speed possible in the universe.

4. Can relativistic speed be greater than the speed of light?

No, according to Einstein's theory of relativity, the speed of light is the maximum speed in the universe. Therefore, relativistic speed cannot exceed the speed of light.

5. How does relativistic speed affect the behavior of objects?

Relativistic speed can have significant effects on the behavior of objects, such as time dilation and length contraction. This means that an object moving at relativistic speeds will experience time passing slower and its length will appear shorter to an outside observer. This is due to the relativistic effects of speed on the fabric of space and time.

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