Question

A proton is accelerated to $0.999c$.

a. What is the proton’s momentum?

b. By what factor does the proton’s momentum exceed its Newtonian momentum?

Short answer

a.) $4.8\times {10}^{-16}kgm/s$

b.) localid="1649819657839" $1000$times less

Step-by-step solution

Part (a) Step 1: Given Information

We have given that a proton is accelerated to $0.999c$.

We have to find the proton’s momentum.

Part (a) Step 2: Simplify

According to relativity the momentum of a particle is given by,

$p=\frac{mv}{1-{\displaystyle \frac{v}{c}}}$

Here,

$m=1.6\times 106-27$and

role="math" localid="1649819364489" $$\begin{gathered} v=0.999C \\ =0.999\times 3.0\times {10}^{8}m/s \end{gathered}$$

So, the momentum is

$$\begin{gathered} p=\frac{1.6\times {10}^{-27}\times 0.999\times 3\times {10}^8}{1-{\displaystyle \frac{0.999C}{C}}} \\ =\frac{4.8\times {10}^{-19}}{0.001} \\ =4.8\times {10}^{-16} \end{gathered}$$

Part (b) Step 1: Given Information

We have given that a proton is accelerated to $0.999c$.

We have to find by what factor does the proton’s momentum exceed its Newtonian momentum.

Part (b) Step 2: Simplify

According to Newtonian mechanics the momentum will be

$$\begin{gathered} p_N=mv \\ =1.6\times {10}^{-27}\times 0.999\times 3\times {10}^8 \\ =4.8\times {10}^{-19} \end{gathered}$$

Therefore, the momentum will be $1000$times less than the relativistic momentum.