Relativistic Kinetic Energy: Does Mass Change?

[Weam Abou Hamdan]

Hello,

According to Special Relativity, the mass of an object must increase as its speed approaches the speed of light.
m=m₀γ
In the formula that allows us to calculate the kinetic energy of a body, KE=0.5mv², should we take into consideration such increase in mass?

Weam Abou Hamdan
Wednesday, August 22, 2018

 

[Nugatory]

Hello,

According to Special Relativity, the mass of an object must increase as its speed approaches the speed of light.
m=m₀γ
In the formula that allows us to calculate the kinetic energy of a body, KE=0.5mv², should we take into consideration such increase in mass?

That $mv^2/2$ formula for the kinetic energy is a classical approximation that only works at speeds that are small compared the speed of light. The correct relativistic expression for the kinetic energy is $(\gamma-1)m_0c^2$ where $\gamma=1/\sqrt{1-\frac{v^2}{c^2}}$ and $m_0$ is the mass of the object when it is at rest. (It would be a good exercise to confirm that this expression approaches $mv^2/2$ when $v$ is small compared with $c$ - expanding $\gamma$ in a power series in $v/c$ will do the trick).

Although you'll hear that bit about the mass of an object increasing a lot, this idea has been largely abandoned in recent decades - there are better ways of describing what happens. This comes up so often that we have an Insights article: https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

 

[Dale]

According to Special Relativity, the mass of an object must increase as its speed approaches the speed of light.

Even as early as Einstein the concept of relativistic mass has been recommended against. Here is a good paper on the topic:

https://arxiv.org/abs/physics/0504110

 

[Sorcerer]

Hello,

According to Special Relativity, the mass of an object must increase as its speed approaches the speed of light.
m=m₀γ
In the formula that allows us to calculate the kinetic energy of a body, KE=0.5mv², should we take into consideration such increase in mass?

Weam Abou Hamdan
Wednesday, August 22, 2018

Please stop thinking about relativistic mass. Instead, find out what proper time is, and use the quantity t/γ as time and do the normal kinematic derivation of kinetic energy. The γ will end up giving you the relativistic kinetic energy formula.

Using t/γ as time:

Divide x by t/γ
Miltiply by m to get relativistic momentum
Take the derivative with respect to t to get force
Integrate over distance (changing the variable to v) to get relativistic kinetic energy.

Basically the same thing you’d do in non-relativistic physics but starting with t/γ instead of t.This old post by a site mentor goes through it more concisely:

https://www.physicsforums.com/threads/derivation-of-relativistic-energy.63380/#post-458653