Relative momentum formula or not?

[leejqs]

Homework Statement

In which of these situations is it reasonable to use the approximate formula for the momentum of an object, instead of the full relativistically correct formula?
A car traveling on the interstate.
A proton in outer space traveling at 2×108 m/s.
A commercial jet plane flying between New York and Seattle.
An electron in a vacuum tube traveling 6×106 m/s.
A neutron traveling at 3600 meters per second.

Homework Equations

Velocities closer to the speed of light require a relativistic formula for momentum.

The Attempt at a Solution

A car traveling on the interstate.
A proton in outer space traveling at 2×108 m/s. (Relativistic)
A commercial jet plane flying between New York and Seattle. (Approximate)
An electron in a vacuum tube traveling 6×106 m/s. (Relativistic)
A neutron traveling at 3600 meters per second. (Approximate)

I am incorrect in these answers... however, what makes a velocity "close" to the speed of light. Would a neutron traveling 3600 m/s require a relativistic formula? 3600 m/s isn't very close to 3E8 m/s.

Any help or insight would be greatly appreciated! Thanks!

 

[Andrew Mason]

The electron in vacuum tube traveling at 6 x 10^6 m/s is traveling at .02c. So the relativistic calculation differs from the approximate by a factor of 1.0002. There is no need to use the relativistic calculation.

To determine whether the relativistic calculation is needed you have to evaluate:

$$\frac{1}{\sqrt{1 - v^2/c^2}}$$

AM

 

[kerimek]

As a scientist, it is important to always use the most accurate and appropriate formula for any given situation. In the case of momentum, it is important to use the relativistically correct formula when dealing with velocities that are close to the speed of light, as the standard formula does not accurately account for the effects of special relativity.

While 3600 m/s may not seem close to the speed of light, it is still a significant fraction of it (about 0.0012c). Therefore, it would be more reasonable to use the relativistic formula for the momentum of a neutron traveling at this speed.

It is also important to note that the choice of formula may depend on the level of precision required for the specific situation. For example, in practical applications such as calculating the momentum of a car on the interstate, the approximate formula may be sufficient. However, for more precise calculations or in situations involving high-energy particles, the relativistic formula should always be used to ensure accuracy.