The total energy of a particle is three times its rest energy. Find $v / c$ for the particle.

What percentage error is made in using $\frac{1}{2} m_0 v^2$ for the kinetic energy of a particle if its speed is 0.90 c$ ?

What percentage error is made in using $\frac{1}{2} m_0 v^2$ for the kinetic energy of a particle if its speed is 0.10 c$?

The rest energy of a proton is about $938 \mathrm{MeV}$. If its kinetic energy is also $938 \mathrm{MeV}$, find its speed.

The rest energy of a proton is about $938 \mathrm{MeV}$. If its kinetic energy is also $938 \mathrm{MeV}$, find its momentum.

Derive the equation $E^2=p^2 c^2+m^2 c^4$ (Equation R-17) by eliminating $v$ from Equations R-10 and R-16.

Use the binomial expansion and Equation R-17 to show that when $p c e r m c^2$, the total energy is given approximately by $E=m c^2+\left(p^2 / 2 m\right)$.

Find the speed of an electron with kinetic energy of $10.0 \mathrm{MeV}$.

Find the speed of an electron with kinetic energy of $0.510 \mathrm{MeV}$.

Show that the speed $v$ of a particle of mass $m$ and total energy $E$ is given by $o / c=\left[1-\left(\left(m c^2\right)^2 / E^2\right)\right]^{1 / 2}$, and that when $E$ is much greater than $m c^2$, this can be approximated by $(v / c)=1-\left(\left(m c^2\right) / 2 E^2\right)$.

Estimation Using a spreadsheet program or graphing calculator, make a graph of the kinetic energy of a particle with rest energy of $100 \mathrm{MeV}$ for speeds between 0 and $c$. On the same graph, plot $\frac{2}{2} m v^2$ by way of comparison. Using the graph, estimate what speed of the nonrelativistic expression is no plot the energy in units of $\mathrm{MeV}$ and the speed in the dimensionless form $v / c$.

What is the total energy of a proton whose momentum is $3 m c$?

If the kinetic energy of a particle equals its rest energy, what percentage error is made by using $p=m v$ for its momentum? Is the nonrelativistic expression always smaller or larger than the relativistically correct expression for momentum?

How much energy would be required to accelerate a particle of mass $m$ from rest to $0.990 c$?