Answer
a. $v=c\sqrt{\frac{3}{4}}=0.866c$.
b. $v=c\sqrt{\frac{5}{9}}=0.745c$.
Work Step by Step
a. The kinetic energy plus the rest energy is equal to the total energy.
The kinetic energy is half the total energy.
We conclude that the kinetic energy equals the rest energy. Use equation 26–5b.
$$KE= (\gamma-1)mc^2= mc^2$$
$$\gamma =2=\frac{1}{\sqrt{1-v^2/c^2}}$$
$$v=c\sqrt{\frac{3}{4}}=0.866c$$
b. Now, we are told that the kinetic energy equals half the rest energy.
$$KE= (\gamma-1)mc^2= \frac{1}{2}mc^2$$
$$\gamma = \frac{3}{2}=\frac{1}{\sqrt{1-v^2/c^2}}$$
$$v=c\sqrt{\frac{5}{9}}=0.745c$$
