At what speed does the classical momentum pmv give an error

At what speed does the classical momentum, p=mv, give an error, when compared with the relativistic momentum, of 2.00 % ?

At what speed does the classical momentum, p=mv, give an error, when compared with the relativistic momentum, of 4.18 % ?

Solution

We compare the classical momentum to the relativistic momentum.

(p_classical /p_relativistic)=mv/(mv/(1-(v/c)²)^1/2

(1-(v/c)²)^1/2=(p_classical /p_relativistic)

error=[1-(p_classical /p_relativistic)]*100

2=[1-(p_classical /p_relativistic)]*100

0.98=(p_classical /p_relativistic)=(1-(v/c)²)^1/2

v=0.199c m/s=0.597*10^8 m/s

b)We compare the classical momentum to the relativistic momentum.

(p_classical /p_relativistic)=mv/(mv/(1-(v/c)²)^1/2

(1-(v/c)²)^1/2=(p_classical /p_relativistic)

error=[1-(p_classical /p_relativistic)]*100

4.18=[1-(p_classical /p_relativistic)]*100

0.9582=(p_classical /p_relativistic)=(1-(v/c)²)^1/2

v=0.2861c m/s=0.8583*10^8 m/s