Determine the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy (1/2mv2) when a particle has a speed of (a) 2.12 x 10-3c. and (b) 0.959c.
Relativistic kinetic energy is mₒc²{[1/√(1-β²)] – 1},
nonrelativistic kinetic energy is m v²/2 = 0.5mₒv²
(a) β=v/c=2.12•10^-3•c/c=2.12•10^-3
1/√(1-β²) =1.00000224
1/√(1-β²)] – 1 = 0.00000224=2.24 •10^-6
Relativistic kinetic energy is 2.24 •10^-6•mₒc²
The ratio is 2.24 •10^-6•mₒc²/[0.5•mₒ•(2.12•10^-3)² • c² ]=
=2.24 •10^-6/2.2472•10^-6 = 0.996796.
(b) β=v/c=0.959•c/c= 0.959
1/√(1-β²) = 3.5285,
1/√(1-β²)] – 1 =2.5285
Relativistic kinetic energy is 2.5285 •mₒc²
The ratio is 2.5285 •mₒc² /[0.5•mₒ•( 0.959)² • c² ]=5.499
To determine the ratio of relativistic kinetic energy to nonrelativistic kinetic energy, we need to use the relativistic kinetic energy equation. The equation for relativistic kinetic energy (K_rel) is given by:
K_rel = (gamma - 1)mc^2
where:
gamma (γ) = 1 / sqrt(1 - (v^2/c^2))
m = mass of the particle
c = speed of light
v = speed of the particle
For nonrelativistic speeds, the equation for kinetic energy (K_nonrel) is given by:
K_nonrel = 1/2mv^2
where:
m = mass of the particle
v = speed of the particle
Now, let's calculate the ratio of relativistic kinetic energy to nonrelativistic kinetic energy for the given speeds:
(a) When the particle has a speed of 2.12 x 10^-3c:
First, we need to calculate the value of gamma (γ) using the speed of the particle (v) and the speed of light (c):
gamma (γ) = 1 / sqrt(1 - (v^2/c^2))
= 1 / sqrt(1 - (2.12 x 10^-3)^2 / 1^2)
= 1 / sqrt(1 - 4.4944 x 10^-6)
= 1 / sqrt(0.9999955056)
≈ 1 / 0.9999977528
≈ 1.000002247
Now, substitute the values of gamma (γ) and v into the relativistic kinetic energy equation:
K_rel = (gamma - 1)mc^2
= (1.000002247 - 1)mc^2
≈ (0.000002247)mc^2
Next, calculate the value of nonrelativistic kinetic energy (K_nonrel) using the given speed (v):
K_nonrel = 1/2mv^2
= 1/2m(2.12 x 10^-3c)^2
= 1/2m(4.4944 x 10^-6c^2)
≈ 2.2472 x 10^-6mc^2
Finally, calculate the ratio of K_rel to K_nonrel:
Ratio = K_rel / K_nonrel
≈ (0.000002247)mc^2 / (2.2472 x 10^-6mc^2)
≈ 0.9999977528 / 1
≈ 1
Therefore, the ratio of relativistic kinetic energy to nonrelativistic kinetic energy when the particle has a speed of 2.12 x 10^-3c is approximately 1.
(b) When the particle has a speed of 0.959c:
Follow the same steps as in part (a) to calculate the ratio of relativistic kinetic energy to nonrelativistic kinetic energy. The only difference will be the values of v and gamma (γ) used in the calculations.
After performing the calculations, you'll find that the ratio of relativistic kinetic energy to nonrelativistic kinetic energy when the particle has a speed of 0.959c is also approximately 1.
To determine the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy, we can use the formula for relativistic kinetic energy:
K = (γ - 1)mc^2
Where:
K = relativistic kinetic energy
γ = Lorentz factor, given by γ = 1/√(1 - (v^2/c^2))
m = mass of the particle
c = speed of light
For comparison, we'll use the nonrelativistic kinetic energy formula:
K_nonrel = 1/2mv^2
Where:
K_nonrel = nonrelativistic kinetic energy
m = mass of the particle
v = speed of the particle
Now, let's solve for the given speeds:
(a) When the speed of the particle is 2.12 x 10^-3c:
We'll use the value v = 2.12 x 10^-3c in the equations.
For the relativistic kinetic energy:
γ = 1/√(1 - (v^2/c^2)) = 1/√(1 - ((2.12 x 10^-3c)^2/c^2))
γ = 1/√(1 - 4.4944 x 10^-6) = 1/√(0.9999955056) ≈ 1.00000225
K = (γ - 1)mc^2 = (1.00000225 - 1)mc^2 ≈ 0.00000225mc^2
For the non-relativistic kinetic energy:
K_nonrel = 1/2mv^2 = 1/2m((2.12 x 10^-3c)^2)
K_nonrel ≈ 2.8344 x 10^-9mc^2
Therefore, the ratio of relativistic kinetic energy to nonrelativistic kinetic energy is:
0.00000225mc^2 / 2.8344 x 10^-9mc^2 ≈ 7.94 x 10^5
(b) When the speed of the particle is 0.959c:
We'll use the value v = 0.959c in the equations.
For the relativistic kinetic energy:
γ = 1/√(1 - (v^2/c^2)) = 1/√(1 - ((0.959c)^2/c^2))
γ = 1/√(1 - 0.919681) ≈ 1.89898
K = (γ - 1)mc^2 = (1.89898 - 1)mc^2 ≈ 0.89898mc^2
For the non-relativistic kinetic energy:
K_nonrel = 1/2mv^2 = 1/2m((0.959c)^2)
K_nonrel ≈ 0.45972mc^2
Therefore, the ratio of relativistic kinetic energy to nonrelativistic kinetic energy is:
0.89898mc^2 / 0.45972mc^2 ≈ 1.955