Question

The mass of a moving electron is 11 times its rest mass. Find its kinetic energy and momentum

Answer 1

Explanation:

First you have to calculate at what velocity will the electron attain 11 times its rest mass, before you will able to calculate its energy and momentum. So, relativistic mass M = m/(1 - v^2/c^2)^1/2, where m is the rest mass, v is the velocity of the mass, and c is the speed of light. Now here M = 11•m as given. So solving for v gives you : v = c(1 - m^2/M^2)^1/2. Hence substituting for M we have v = c(1 - m^2/121m^2)^1/2 which simplifies to v = c(1 - 1/121)^1/2 which is equal to 0.991735548c, or 2.975 • 10^8m/s. Electron's 'rest' mass is 9.11 • 10^-31 kg. So the KE is 1/2mv^ = 1/2(11 • 9.11 • 10^-31 kg)(2.975 • 10^8 m/s)^2= 443.461 • 10^-15 J or 4.4346 • 10^-13 J(Joules). And the momentum is p = mv = (11• 9.11 • 10^-31kg)(2.975•10^8m/s) = 298.125 • 10^-23 kg•m/s or 2.9813 • 10^-21 kg•m/s . The above are calculated values of the relativistic kinetic energy, and relativistic momentum for an electron traveling at 2.975 • 10^8m/s, bearing 11 times its rest mass as a result of its speed…

Answer 2

Given:

mass of a moving electron is 11 times its rest mass

To find:

Its kinetic energy and momentum

Solution:

Let the moving mass of the electron be $m$ and resting mass be $m_0$.

$m=11m_0$

So, we calculate kinetic energy as,

$K.E.=(m-m_0)c^2$

$=(11-1)m_0c^2$

Now we know the resting mass of an electron $m_0=9.1$×$10^{-31}$kg and speed of light $c=3$×$10^8$m/s

So,

$K.E.=10$×$9.1$×$10^{-31}$×$(3$×$10^8)^2$

$=8.19$×$10^{-13}$ J

Now, we know that $momentum=mass$×$velocity$

By using mass variation formula we can say that,

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2} } }$

Putting $m=11m_0$, we solve for v,

$11m_0=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2} } }$

⇒$\frac{1}{11}=\sqrt{1-\frac{v^2}{c^2} }$

⇒$\frac{v^2}{c^2}=1-\frac{1}{121}$

⇒$\frac{v}{c} =\sqrt{\frac{120}{121} }$

⇒$\frac{v}{c} =0.995$

⇒$v=0.995$×$3$×$10^8$

⇒$v=2.985$×$10^8$ m/s

Now we can calculate the momentum

$b=11m_0$×$2.985$×$10^8$

$=2.987$×$10^{-21}$ N-s

Hence, the Kinetic energy is $8.19$×$10^{-13}$ J and the momentum is $2.987$×$10^{-21}$ N-s.