Find the kinetic energy of a particle with a mass of one gram moving with half the speed of light. Compare the answer with the one you would find using the nonrelativistic formula. Ans in units of x 10^13.
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html
To find the kinetic energy of a particle moving with a significant fraction of the speed of light, we need to use the relativistic formula for kinetic energy:
K = (γ - 1) * (mc^2)
Where:
K = relativistic kinetic energy
γ (gamma) = Lorentz factor
m = mass of the particle
c = speed of light in a vacuum
In this case, the mass of the particle is given as one gram (1 g), and the speed is half the speed of light (c/2). We can first convert the mass to kilograms for consistency:
1 gram = 0.001 kilograms
Now we can find the value of the Lorentz factor using the formula:
γ = 1 / sqrt(1 - (v^2 / c^2))
Where:
v = velocity of the particle
In this case, the velocity is given as half the speed of light (c/2). Substituting the values into the formula:
γ = 1 / sqrt(1 - ((c/2)^2 / c^2))
= 1 / sqrt(1 - (1/4))
= 1 / sqrt(3/4)
= 1 / (√3 / 2)
= 2 / √3
To find the relativistic kinetic energy, we substitute the values of γ, m, and c into the formula:
K = (γ - 1) * (mc^2)
= (2 / √3 - 1) * (0.001 * (3 x 10^8)^2)
Evaluating the expression:
K = (2 / √3 - 1) * 0.001 * (9 x 10^16)
= (2 / √3 - 1) * 9 x 10^13
Now we can compare this with the nonrelativistic formula for kinetic energy:
K_nonrel = (1/2) * mv^2
Substituting the values of m and v:
K_nonrel = (1/2) * 0.001 * ((3 x 10^8) / 2)^2
Calculating:
K_nonrel = 0.001 * (225 x 10^16) / 4
= 0.001 * 56.25 x 10^16
= 56.25 x 10^13
So, the relativistic kinetic energy is (2 / √3 - 1) * 9 x 10^13, while the nonrelativistic kinetic energy is 56.25 x 10^13.
Note that these values are multiples of 10^13 units, as requested.