an electron has a kinetic energy that is 1.8 times its rest energy. What is its speed, expressed as a fraction of the speed of light?

To calculate the speed of an electron in terms of the speed of light, we need to know its kinetic energy and rest energy. The rest energy of an electron is given by the equation:

Erest = mc^2

where Erest is the rest energy, m is the mass of the electron, and c is the speed of light.

Given that the electron's kinetic energy is 1.8 times its rest energy, we can write:

Ekinetic = 1.8Erest

Now, the total energy of the electron is the sum of its rest energy and kinetic energy:

Etotal = Erest + Ekinetic

Since the total energy of the electron is equivalent to its kinetic energy due to its high velocity, we can write:

Etotal = Ekinetic

Now, substituting the equation for kinetic energy:

Etotal = 1.8Erest

Next, we can equate the two expressions for the total energy:

Etotal = mc^2

Substituting the value of kinetic energy:

1.8Erest = mc^2

Dividing both sides of the equation by c^2:

(1.8Erest) / c^2 = m

Since we know that the kinetic energy is given by:

Ekinetic = (1/2)mv^2

where v represents the velocity of the electron, we can rewrite the equation as:

(1.8Erest) / c^2 = (1/2)mv^2

Now, we can substitute the value of the rest energy using the equation:

Erest = mc^2

This gives us:

(1.8(mc^2)) / c^2 = (1/2)mv^2

Simplifying the equation:

1.8m = (1/2)mv^2

Dividing both sides of the equation by m:

1.8 = (1/2)v^2

Multiplying both sides of the equation by 2:

3.6 = v^2

Taking the square root of both sides:

v = √3.6

So, the speed of the electron, expressed as a fraction of the speed of light (c), is √3.6 / c.

m•c^2 = 1.8•m(o) •c^2,

m(o) •c^2/sqrt(1- β^2) = 1.8•m(o) •c^2,
sqrt(1- β^2) =1/1.8
β = 0.831
β = v/c ,
v = 0.831c = 2.49•10^8 m/s