Determine the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy (1/2mv2) when a particle has a speed of (a) 2.60 × 10-3c. and (b) 0.972c
To determine the ratio of relativistic kinetic energy to nonrelativistic kinetic energy, we need to use the relativistic expression for kinetic energy.
The relativistic kinetic energy (K) of a particle with mass (m) and velocity (v) can be calculated using the formula:
K = (γ - 1)mc²
where γ is the Lorentz factor given by:
γ = 1 / √(1 - (v²/c²))
Here, c is the speed of light.
(a) For a particle with a speed of 2.60 × 10^-3c, we can substitute the values into the formula to find the ratio.
First, let's calculate the Lorentz factor γ:
γ = 1 / √(1 - (v²/c²))
= 1 / √(1 - ((2.60 × 10^-3c)²/c²))
Simplifying the equation further, we have:
γ = 1 / √(1 - 6.76 × 10^-6)
To find the value of γ, we can take the square root of (1 - 6.76 × 10^-6):
γ ≈ 1.0000000000338
Next, we can calculate the relativistic kinetic energy (K) using the formula:
K = (γ - 1)mc²
= (1.0000000000338 - 1)mc²
Since the nonrelativistic kinetic energy is given by (1/2)mv², the required ratio is:
Ratio = K / (1/2)mv²
Substituting the values, we get:
Ratio = ((1.0000000000338 - 1)mc²) / [(1/2)mc²v²]
= (0.0000000000338mc²) / [(1/2)mc²v²]
= 0.0000000000676v²
Therefore, the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy for a particle with a speed of 2.60 × 10^-3c is approximately 0.0000000000676v².
(b) For a particle with a speed of 0.972c, repeat the same calculations to find the ratio:
First, calculate the Lorentz factor γ:
γ = 1 / √(1 - (v²/c²))
= 1 / √(1 - (0.972c)²/c²)
Simplifying the equation further, we have:
γ = 1 / √(1 - 0.945984c²/c²)
= 1 / √(0.054016)
Taking the square root of 0.054016:
γ ≈ 3.285696
Next, calculate the relativistic kinetic energy (K):
K = (γ - 1)mc²
= (3.285696 - 1)mc²
= 2.285696mc²
Substituting this value into the ratio:
Ratio = K / (1/2)mv²
= (2.285696mc²) / [(1/2)mc²v²]
= 4.571392 / v²
Therefore, the ratio of the relativistic kinetic energy to the nonrelativistic kinetic energy for a particle with a speed of 0.972c is approximately 4.571392 / v².
To determine the ratio of relativistic kinetic energy to nonrelativistic kinetic energy, we need to use the relativistic formula for kinetic energy.
The relativistic formula for kinetic energy is given by:
K = (γ - 1)mc²
where K is the kinetic energy, γ is the Lorentz factor, m is the mass of the particle, and c is the speed of light in a vacuum.
In nonrelativistic situations, the Lorentz factor (γ) is approximately equal to 1, so we can simplify the relativistic formula for kinetic energy to the nonrelativistic formula:
K ≈ 1/2mv²
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 2.60 × 10^(-3)c:
To begin, we need to calculate the Lorentz factor (γ) using the formula:
γ = 1 / √(1 - (v/c)²)
Given that v = 2.60 × 10^(-3)c, we can substitute this value into the formula:
γ = 1 / √(1 - (2.60 × 10^(-3)c / c)²)
= 1 / √(1 - (2.60 × 10^(-3))²)
= 1 / √(1 - 6.76 × 10^(-6))
≈ 1 / √(1 - 0)
≈ 1
Now let's calculate the relativistic kinetic energy (K_rel) and the nonrelativistic kinetic energy (K_nonrel) using their respective formulas:
K_rel = (γ - 1)mc²
= (1 - 1)mc²
= 0
K_nonrel = 1/2mv²
The ratio of relativistic kinetic energy to nonrelativistic kinetic energy is:
K_rel / K_nonrel
= 0 / (1/2mv²)
= 0
Therefore, the ratio of relativistic kinetic energy to nonrelativistic kinetic energy is 0 when the particle has a speed of 2.60 × 10^(-3)c.
(b) When the particle has a speed of 0.972c:
Similarly, we need to calculate the Lorentz factor (γ) using the formula:
γ = 1 / √(1 - (v/c)²)
Given that v = 0.972c, we can substitute this value into the formula:
γ = 1 / √(1 - (0.972c / c)²)
= 1 / √(1 - 0.945²)
= 1 / √(1 - 0.892)
= 1 / √(0.108)
= 1 / 0.328
≈ 3.05
Now let's calculate the relativistic kinetic energy (K_rel) and the nonrelativistic kinetic energy (K_nonrel) using their respective formulas:
K_rel = (γ - 1)mc²
= (3.05 - 1)mc²
= 2.05mc²
K_nonrel = 1/2mv²
The ratio of relativistic kinetic energy to nonrelativistic kinetic energy is:
K_rel / K_nonrel
= (2.05mc²) / (1/2mv²)
= 4.1c² / v²
Since the value of c is the speed of light and constant, we can write the ratio as:
4.1 / v²
Substituting v = 0.972c:
Ratio = 4.1 / (0.972c)²
= 4.1 / 0.945
≈ 4.34
Therefore, the ratio of relativistic kinetic energy to nonrelativistic kinetic energy is approximately 4.34 when the particle has a speed of 0.972c.