An x-ray tube is an evacuated glass tube that produces electrons at one end and then accelerates them to very high speeds by the time they reach the other end. The acceleration is accomplished using an electric field. The high-speed electrons hit a metal target at the other end, and the violence of the collision converts their kinetic energy into high-energy light rays, commonly known as x-rays. Required:
a. Through what potential difference should electrons be accelerated so that their speed is 2.7 % of the speed of light when they hit the target?
b. What potential difference would be needed to give protons the same kinetic energy as the electrons?
c. What speed would this potential difference give to protons?

a)  V = 1.866 10² V ,  b)   V = 3.424 10⁵ V , c)   v = 8.1 10⁶ m / s

Explanation:

a) the potential difference is requested to accelerate the electrons up to 2.7% of the speed of light

           v = 0.027 c

           v = 0.027 3 10⁸

           v = 8.1 10⁶ m / s

for this part we can use the conservation of mechanical energy

starting point. When electrons are at rest

           Em₀ = U = q V

final point. Electrons with maximum speed

          Em_f = K = ½ m v2

          Em₀ = Em_{f}

          e V = ½ m v²

          V = ½ m v² / e

let's calculate

          V = ½  9.1 10⁻³¹ (8.1 10⁶)² / 1.6 10⁻¹⁹

          V = 1.866 10² V

           V = 1866 V

b) if this acceleration protons is the mass of the proton is m_{p} = 1.67 10-27

          V = ½ 1.67 10⁻²⁷ (8.1 10⁶)² / 1.6 10⁻¹⁹

           V = 3.424 10⁵ V

           V = 342402 V

c)   this potential difference should give the protons the same speed as the electrons

             v = 8.1 10⁶ m / s

think its the third option