4. In experiments, physicists routinely accelerate protons to speeds quite close to the speed of light. The mass of a proton is $1.67 \times 10^{- 27} kg$ , 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.

Answer

Expert–verified

Hide Steps

Answer

$\frac{p_{r e l}}{p} \approx 7.09$

Steps

Step 1 ：Given the mass of the proton (m) as $1.67 \times 10^{- 27} kg$ and its speed (v) as $0.99 c$ , 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 $p_{r e l} = \frac{m \times v}{\sqrt{1 - (\frac{v}{c})^{2}}}$ .

Step 4 ：Determine the ratio between the relativistic momentum and the Newtonian momentum as $\frac{p_{r e l}}{p}$ .

Step 5 ： $p = 4.96 \times 10^{- 19} kg m / s$

Step 6 ： $p_{r e l} = 3.52 \times 10^{- 18} kg m / s$

Step 7 ： $\frac{p_{r e l}}{p} \approx 7.09$