4. In experiments, physicists routinely accelerate protons to speeds quite close to the speed of light. The mass of a proton is $1.67 \times 10^{-27} \mathrm{~kg}$, and the proton is moving with a speed of $0.99 c$. (a) Calculate the proton's momentum, according to Newton's definition. (b) Calculate the proton's relativistic momentum. (c) Determine the ratio of the relativistic momentum to the Newtonian momentum.

Step 1 ：Given the mass of the proton (m) as \(1.67 \times 10^{-27} \mathrm{~kg}\) and its speed (v) as \(0.99c\), where c is the speed of light.

Step 2 ：Calculate the Newtonian momentum (p) using the formula \(p = m \times v\).

Step 3 ：Calculate the relativistic momentum (p_{rel}) using the formula \(p_{rel} = \frac{m \times v}{\sqrt{1 - (\frac{v}{c})^2}}\).

Step 4 ：Determine the ratio between the relativistic momentum and the Newtonian momentum as \(\frac{p_{rel}}{p}\).

Step 5 ：\(p = 4.96 \times 10^{-19} \mathrm{~kg~m/s}\)

Step 6 ：\(p_{rel} = 3.52 \times 10^{-18} \mathrm{~kg~m/s}\)

Step 7 ：\(\boxed{\frac{p_{rel}}{p} \approx 7.09}\)