Show that the kinetic energy K of a particle of mass m is related to its momentum p by the equation $p=\frac{\sqrt{K^2+2Km{c}^2}}{c}$.

The Sun radiates energy at a rate of about 4 x 10²⁶ W.

(a) At what rate is the Sun’s mass decreasing?

(b) How long does it take for the Sun to lose a mass equal to that of Earth?

(c) Estimate how long the Sun could last if it radiated constantly at this rate.

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

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process. See Section 37–5.)

An electron (m = 9.11 x 10⁻³¹ kg) is accelerated from rest to speed v by a conservative force. In this process, its potential energy decreases by 7.20 x 10⁻¹⁴ J . Determine the electron’s speed, v.

A spaceship and its occupants have a total mass of 160,000 kg. The occupants would like to travel to a star that is 32 light-years away at a speed of 0.70c. To accelerate, the engine of the spaceship changes mass directly to energy.

(a) Estimate how much mass will be converted to energy to accelerate the spaceship to this speed.

(b) Assuming the acceleration is rapid, so the speed for the entire trip can be taken to be 0.70c, determine how long the trip will take according to the astronauts on board.

In one of Thomson’s experiments he placed a thin metal foil in the electron beam and measured its temperature rise. Consider a cathode-ray tube in which electrons are accelerated through a 2000 V potential difference, then strike a 10 mg copper foil. What is the electron-beam current if the foil temperature rises 6.0°C in 10 s? Assume no loss of energy by radiation or other means. The specific heat of copper is 385 J/kg K .

You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. By how much? Hint: Use the binomial approximation.

Electrons pass through the parallel electrodes shown in FIGURE EX37.9 with a speed of 5.0×10⁶ m/s. What magnetic field strength and direction will allow the electrons to pass through without being deflected? Assume that the magnetic field is confined to the region between the electrodes.