If $E$ is the total energy of a particle with zero potential energy, show that $d E / d p=v$, where $p$ and $v$ are the momentum and velocity of the particle, respectively.

As a rough rule, anything traveling faster than about $0.1 c$ is called relativistic - that is, special relativity is a significant effect. Determine the speed of an electron in a hydrogen atom (radius $0.53 \times 10^{-10} \mathrm{~m}$ ) and state whether or not it is relativistic. (Treat the electron as though it were in a circular orbit around the proton.)

$(a)$ What is the speed $v$ of an electron whose kinetic energy is $14,000$ times its rest energy? You can state the answer as the difference $c-v$. Such speeds have been reached in the Stanford Linear Accelerator, SLAC. $(b)$ If the electrons travel in the lab through a tube $3.0 \mathrm{~km}$ long (as at SLAC), how long is this tube in the electrons' reference frame? [Hint: Use the binomial expansion.]

Rocket $A$ passes Earth at a speed of $0.65 c$. At the same time, rocket $B$ passes Earth moving $0.90 \mathrm{c}$ relative to Earth in the same direction. How fast is $B$ moving relative to $A$ when it passes $A$ ?

What is the speed $v$ of an electron whose kinetic energy is $14,000$ times its rest energy? You can state the answer as the difference $c-v$. Such speeds have been reached in the Stanford Linear Accelerator, SLAC.

Rocket A passes Earth at a speed of 0.65 c. At the same time, rocket B passes Earth moving 0.85 c relative to Earth in the same direction. How fast is B moving relative to A when it passes A?

What is the speed v of an electron whose kinetic energy is 14,000 times its rest energy? You can state the answer as the difference $c-v$. Such speeds are reached in the Stanford Linear Accelerator, SLAC.

A spacecraft (reference frame S') moves past Earth (reference frame S) at velocity $\overrightarrow{\mathbf{v}}$, which points along the x and x' axes. The spacecraft emits a light beam (speed c) along its y' axis as shown in the given figure. Compare these relativistic results to what you would have obtained classically (Galilean transformations).

A healthy astronaut's heart rate is 60 beats/min. Flight doctors on Earth can monitor an astronaut's vital signs remotely while in flight. How fast would an astronaut be flying away from Earth if the doctor measured her heart rate as $52 ~beats / \mathrm{min}$ ?

A spacecraft (reference frame S') moves past Earth (reference frame S) at velocity $\overrightarrow{\mathbf{v}}$, which points along the x and x' axes. The spacecraft emits a light beam (speed c) along its y' axis as shown in the given figure. Use velocity transformations to show that the light moves with speed c also in the Earth's reference frame.

At what speed $v$ will the length of a $1.00~\mathrm{m}$ stick look $10.0 \%$ shorter $(90.0 \mathrm{~cm})$ ?

A spacecraft (reference frame $\mathrm{S}^{\prime}$ ) moves past Earth (reference frame S) at velocity $\overrightarrow{\mathbf{v}}$, which points along the $x$ and $x^{\prime}$ axes. The spacecraft emits a light beam (speed $c$ ) along its $y^{\prime}$ axis as shown in Fig. 36-17. (a) What angle $\theta$ does this light beam make with the $x$ axis in the Earth's reference frame?

Does time dilation mean that time actually passes more slowly in moving reference frames or that it only seems to pass more slowly?

A healthy astronaut's heart rate is 60 beats/min. Flight doctors on Earth can monitor an astronaut's vital signs remotely while in flight. How fast would an astronaut have to be flying away from Earth in order for the doctor to measure her having a heart rate of 30 beats/min?