If the total energy of particle is exactly three times of its rest energy.What is the velocity of particle
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If the total energy of a particle is exactly 4/5 of its rest energy, what is the velocity of a particle?
Total energy of a particle or a mass is equal to its rest energy and kinetic energy. So the above question is incorrect ! Total energy can never be less than its rest energy !! The lowest total energy will be equal to the particle’s rest energy when the particle has zero kinetic energy, in other words not in motion. What the asker is likely trying to ask is if the total energy is 5/4 of the particle’s rest energy ?! This could be calculated with the relativistic mass equation which is : M = m/(1 - v^2/c^2)^1/2. Here M is the relativistic mass, m is the rest mass, v is the object’s speed, and c is the speed of light. Let’s say the relativistic mass of a particle m is 5/4m, at velocity v, meaning more than its rest energy due to the added kinetic energy. Then let’s set up the equation : 5/4m = m/(1 - v^2/c^2)^1/2. As you can see m will cancel out from both sides. This will give you : 5/4 = 1/(1 - v^2/c^2)^1/2, which will give you the following : (1 - v^2/c^2)^1/2 = 4/5, which will give you (1 - v^2/c^2) = 16/25, which simplifies to v^2/c^2 = 1 - 16/25, which gives you v^2 = 0.36c^2, which will give you v = 0.6c, or 1.8 • 10^8 m/s. So in essence for a particle or object of any mass to increase its total energy by a 1/4 of its rest energy it will have to travel at 3/5 th the speed of light ! Kaiser T, MD.