Question:

A particle accelerator is used to accelerate protons to a speed of 0.8c, where c is the speed of light. The protons then collide with each other, creating a spray of subatomic particles. One of the resultant particles has a rest mass of 3.6 x 10^-27 kg and a kinetic energy of 2.0 x 10^-16 J.

a) Calculate the relativistic kinetic energy of the particle.

b) Determine the momentum of the particle.

c) Calculate the speed of the particle.

Useful constants: Rest mass of a proton, m = 1.67 x 10^-27 kg Speed of light, c = 3.00 x 10^8 m/s

Answer:

a) The relativistic kinetic energy (K) is given by the equation: [ K = (\gamma - 1)mc^2 ] where: $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

First, calculate the value of $\gamma$: [ \frac{v}{c} = 0.8 ] [ \frac{v^2}{c^2} = (0.8)^2 = 0.64 ] [ 1 - \frac{v^2}{c^2} = 1 - 0.64 = 0.36 ] [ \sqrt{1 - \frac{v^2}{c^2}} = \sqrt{0.36} = 0.6 ]

So, $\gamma = \frac{1}{0.6} = \frac{5}{3}$

Now, substitute the values into the relativistic kinetic energy equation: [ K = \left(\frac{5}{3} - 1\right) \times 3.6 \times 10^{-27} \times (3.00 \times 10^8)^2 ] [ K = \left(\frac{2}{3}\right) \times 3.6 \times 10^{-27} \times 9.00 \times 10^{16} ] [ K = 2.4 \times 3.6 \times 9.00 \times 10^{-11} ] [ K = 7.776 \times 10^{-10} J ]

Therefore, the relativistic kinetic energy of the particle is 7.776 x 10^-10 J.

b) The momentum (p) of the particle can be calculated using the equation: [ p = \gamma mv ] [ p = \frac{5}{3} \times 3.6 \times 10^{-27} \times 0.8 \times 3.00 \times 10^8 ] [ p = \frac{5}{3} \times 3.6 \times 0.80 \times 3.00 \times 10^{-19} ] [ p = 0.8 \times 3 \times 3 \times 10^{-19} ] [ p = 7.2 \times 10^{-19} , \text{kg m/s} ]

Therefore, the momentum of the particle is 7.2 x 10^-19 kg m/s.

c) The speed of the particle can be determined using the momentum and the relativistic equation: [ v = \frac{p}{\gamma m} ] [ v = \frac{7.2 \times 10^{-19}}{\frac{5}{3} \times 3.6 \times 10^{-27}} ] [ v = \frac{7.2 \times 10^{-19}}{\frac{5}{3} \times 3.6 \times 10^{-27}} ] [ v = \frac{7.2 \times 3}{5 \times 3.6} \times 10^{8} ] [ v = 1.2 \times 10^{8} ]

Therefore, the speed of the particle is 1.2 x 10^8 m/s.

In summary: a) The relativistic kinetic energy of the particle is 7.776 x 10^-10 J. b) The momentum of the particle is 7.2 x 10^-19 kg m/s. c) The speed of the particle is 1.2 x 10^8 m/s.