Question

At what speed does the momentum of a particle differ by 2.00% from the value obtained with the nonrelativistic expression mv?

Answer 1

A) At what speed does the momentum of a particle differ?by 1% from the value obtained wit the non-relativistic expression mv? I can already answer B. It should be larger because the relativistic equation is:

p = mv / sqrt(1 - v^2/c^2)
and the quantity 'gamma' which is 1 / sqrt(1 - v^2/c^2)
is always greater than 1 unless the object isn't moving in which case it equals 1 so you will be multiplying mv by a number larger than 1 which will make the relativistic momentum bigger.

Now for part A.
To find % difference you subtract the actual value and the other value, and divide by the actual value.
In our case the actual value should be relativistic and we are comparing it to the normal momentum equation so:
? is gamma
(| ?mv - mv | / ?mv) * 100% = 1%
since ?mv is larger we can drop the absolute value
(mv(?-1) / ?mv) * 100% = 1%
((?-1)/?) * 100% = 1%
1 - 1/? = 1/100
99/100 = 1/?
1/? = 99/100

sqrt(1 - v^2/c^2) = 99/100

v = 4.23 x10 ^ 7 m/s
which surprisingly is 14% the speed of light O.o