91Ó°ÊÓ

An unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is equal to \(2.50 \times 10^{-28} \mathrm{~kg}\) and that of the heavier fragment is \(1.67 \times 10^{-27} \mathrm{~kg} .\) If the lighter fragment has a speed of \(0.893 c\) after the breakup, what is the speed of the heavier fragment?

A proton moves with a speed of \(0.950 c\). Calculate (a) its rest energy, (b) its total energy, and (c) its kinetic energy.

Superman throws a \(0.15\) -kg baseball at a speed of \(0.900 c\). Find the baseball's (a) rest energy, (b) kinetic energy, and (c) total energy.

What speed must a particle attain before its kinetic energy is double the value predicted by the nonrelativistic expression \(K E=\frac{1}{2} m v^{2} ?\)

Determine the energy required to accelerate an electron from (a) \(0.500 c\) to \(0.900 c\) and (b) \(0.900 c\) to \(0.990 c\).

A cube of steel has a volume of \(1.00 \mathrm{~cm}^{3}\) and a mass of \(8.00 \mathrm{~g}\) when at rest on Earth. If this cube is now given a speed \(v=0.900 c\), what is its density as measured by a stationary observer? Note that relativistic density is \(E_{R} / c^{2} V\).

An unstable particle with a mass equal to \(3.34 \times 10^{-27} \mathrm{~kg}\) is initially at rest. The particle decays into two fragments that fly off with velocities of \(0.987 c\) and \(-0.868 c\), respectively. Find the masses of the fragments. Hint: Conserve both mass-energy and momentum.

Starting with the definitions of relativistic energy and momentum, show that \(E^{2}=p^{2} c^{2}+m^{2} c^{4}(\) Eq. \(26.10\) )

Consider electrons accelerated to a total energy of \(20.0 \mathrm{GeV}\) in the \(3.00\) -km-long Stanford Linear Accelerator. (a) What is the \(\gamma\) factor for the electrons? (b) How long does the accelerator appear to the electrons? Electron mass energy: \(0.511 \mathrm{MeV}\).

An electron has a speed of \(0.750 c\). (a) Find the speed of a proton that has the same kinetic energy as the electron. (b) Find the speed of a proton that has the same momentum as the clectron.