Electron energy and mass given, find velocity

[JonNash]

Homework Statement

For an electron having energy 1.5MeV/c (mass of an electron is 0.5MeV), the velocity is given by?
a) 2.92x10¹⁰ cm/sec
b) 2.81x10¹⁰ cm/sec
c) 2.75x10¹⁰ cm/sec
d) 2.50x10¹⁰ cm/sec

Homework Equations

Kinetic Energy = 0.5mv²

The Attempt at a Solution

I first converted energy into joules and mass into kg. E=2.4032x10^-13 and mass = 8.91331x10^-31. Then I put it into the equation for kinetic energy and solved for v which yields the result v = 7.3432x10¹⁰ cm/s which is not among the choices. Then I observed that energy is not given in simply eV, at the end there is MeV/c. I am unable to understand what this means. It would unbalance the units on RHS. Is there another formula to be used instead of KE and what does MeV/c mean?

 

[hilbert2]

The electron is moving at a relativistic speed, you can't use the Newtonian kinetic energy formula.

In special relativity, the total energy of a particle is ##E=\frac{m_{0}c^{2}}{\sqrt{1-v^{2}/c^{2}}}##, where ##m_{0}## is the rest mass of the particle and ##v## is the velocity. From this you can solve the velocity when the total energy is known.

 

[vanhees71]

It's not so clear what's given. It depends on how you define what's meant with
"an electron having energy 1.5MeV" (btw. it must be MeV if it's supposed to be an energy, MeV/c is the unit for a momentum). It could mean that the electron has a kinetic energy of 1.5 MeV. Then you have to use
[tex]E_{\text{kin}}=m_0 c^2 \left ( \frac{1}{\sqrt{1-v^2/c^2}}-1 \right )[/tex]
or it could mean it has relativistic total energy, including the rest energy, i.e.,
[tex]E=\frac{m_0 c^2}{\sqrt{1-v^2/c^2}}.[/tex]
So the first thing is to check, which of the quantities is really given.

 

[JonNash]

The electron is moving at a relativistic speed, you can't use the Newtonian kinetic energy formula.

In special relativity, the total energy of a particle is ##E=\frac{m_{0}c^{2}}{\sqrt{1-v^{2}/c^{2}}}##, where ##m_{0}## is the rest mass of the particle and ##v## is the velocity. From this you can solve the velocity when the total energy is known.

I solved it with the formula you gave and I got a complex number
v=√-81x10⁵²

It's not so clear what's given. It depends on how you define what's meant with
"an electron having energy 1.5MeV" (btw. it must be MeV if it's supposed to be an energy, MeV/c is the unit for a momentum). It could mean that the electron has a kinetic energy of 1.5 MeV. Then you have to use
[tex]E_{\text{kin}}=m_0 c^2 \left ( \frac{1}{\sqrt{1-v^2/c^2}}-1 \right )[/tex]
or it could mean it has relativistic total energy, including the rest energy, i.e.,
[tex]E=\frac{m_0 c^2}{\sqrt{1-v^2/c^2}}.[/tex]
So the first thing is to check, which of the quantities is really given.

I solved it with kinetic energy too and I got v=0. Ugghh...

 

[hilbert2]

From the formula of total energy, you can solve that the elctron speed is

##v=c\sqrt{1-\frac{m_{0}^{2}c^{4}}{E^{2}}}## .

There's no way how this could give a complex-valued speed. The total energy ##E## is always larger than or equal to the rest mass energy ##m_ {0}c^{2}##, therefore the number inside the square root must be positive.

Also note that the electron mass is given in natural units, where it has dimensions of energy. When we say "##m_{0}=0.5## MeV" we actually mean "##m_{0}c^{2}=0.5## Mev".

 

[JonNash]

Also note that the electron mass is given in natural units, where it has dimensions of energy. When we say "##m_{0}=0.5## MeV" we actually mean "##m_{0}c^{2}=0.5## Mev".

Ohkay. Mass = m₀c² = 0.5MeV. Armed with this when I solved I got √8x10¹⁰cm/s as the velocity which translates to 2.828x10¹⁰cm/s which closely matches option b). Thanks hilbert2.

 

[RMM]

The root of 8*19^10 cm is 2,828E+05 It is some years ago. probably everybody knows already. I just want to be sure about this. It does not match any of the given values.