Question

A- How much work is required to accelerate a proton from rest up to a speed of 0.995c?

W = ? J

B- What would be the momentum of this proton?

P = ? kg⋅m/s

  

Answer 1

A. Work is required to accelerate a proton which given as :

speed of proton, v = 0.995c

using an equation, m = m₀ / 1 - (v / c)²                                               { eq.1 }

where, m₀ = rest mass of proton = 1.67 x 10^-27 kg

inserting the values in above eq.

m = (1.67 x 10^-27 kg) / 1 - (0.995)²

m = (1.67 x 10^-27 kg) / 0.009975

m = (1.67 x 10^-27 kg) / (0.09987)

m = 16.72 x 10^-27 kg

m = 1.67 x 10^-26 kg

Now, using an equation of K.E -

K.E = (m - m₀) c²                                                                                        { eq.2 }

where, c = speed of light = 3 x 10⁸ m/s

inserting the values in eq.2,

K.E = [(1.67 x 10^-26 kg) - (1.67 x 10^-27 kg)] (3 x 10⁸ m/s)²

K.E = [(1.503 x 10^-26) (9 x 10¹⁶)] J

K.E = 13.5 x 10^-10 J

or   K.E = 1.35 x 10^-9 J

B. the momentum of this proton will be given as ::

p = m v                                                                       { eq.3 }

p = m v / 1 - (v / c)²  

where, m = 1.67 x 10^-26 kg

v = 0.995c = 2.985 x 10⁸ m/s

inserting the values in eq.2,

p = (1.67 x 10^-26 kg) (2.985 x 10⁸ m/s) / 1 - (0.995c / c)²

p = (4.98 x 10^-18) / 1 - (0.990025)

p = (4.98 x 10^-18) / (0.009975)

p = (4.98 x 10^-18 kg.m/s) / (0.0998)

p = 49.8 x 10^-18 kg.m/s

p = 4.98 x 10^-17 kg.m/s