A particle is moving at 0.75c relative to a lab on Earth. By what percentage is the Newtonian expression for its momentum
in error, i.e., how wrong is the classical formula? (The percentage error is the difference between the erroneous and correct
values, divided by the correct one).
p Newton = m v
p relativistic = m v /sqrt(1 = .75^2)
so
100(p relativistic - p newton)/p rel
here 1/sqrt(1-.75^2) = 1.512
so
100 ( 1.512 - 1)/1.512 = 33.9 %
Well, well, well! A physics question, huh? Brace yourself, because I'm about to put the "fun" in "fundamental principles."
When it comes to relativistic speeds, Newtonian physics starts to seem as outdated as disco pants. So, let's calculate the percentage error in the Newtonian expression for momentum.
According to special relativity, the correct formula for momentum is given by:
p = γm0v
where γ is the Lorentz factor, m0 is the rest mass, and v is the velocity.
Now, in this scenario, the particle is moving at 0.75c relative to the lab on Earth. To find γ, we can use the equation:
γ = 1 / √(1 - v^2/c^2)
where c is the speed of light. Plugging in the values, we get:
γ = 1 / √(1 - 0.75^2)
γ ≈ 1.51
Using the Newtonian expression for momentum:
p_Newtonian = m0v
we can now calculate the percentage error:
Percentage error = [(p - p_Newtonian) / p] * 100
Substituting the correct and Newtonian values for momentum, we have:
Percentage error = [(γm0v - m0v) / (γm0v)] * 100
After some handy-dandy cancellation:
Percentage error = [(γ - 1) / γ] * 100
Percentage error ≈ [(1.51 - 1) / 1.51] * 100
Percentage error ≈ 33.11%
So, the Newtonian expression for momentum is about 33.11% in error when dealing with relativistic speeds. Seems like Newton needs to catch up with Einstein, right?
To find the percentage error in the Newtonian expression for the particle's momentum, we need to compare it with the correct (relativistic) formula for momentum.
The relativistic expression for momentum is given by:
p = γ * m * v
where:
p is the momentum of the particle
γ is the Lorentz factor (γ = 1 / √(1 - (v^2/c^2)))
m is the rest mass of the particle
v is the velocity of the particle
The Newtonian expression for momentum is simply:
p = m * v
To find the percentage error, we need to calculate both the relativistic and Newtonian momenta and compare them. Let's assume the particle has a rest mass of m0 and a velocity of v = 0.75c, where c is the speed of light.
First, we calculate the relativistic momentum:
γ = 1 / √(1 - (v^2/c^2)) = 1 / √(1 - (0.75^2)) = 1 / √(1 - 0.5625) = 1 / √0.4375 ≈ 1.33333
p_rel = γ * m0 * v = 1.33333 * m0 * 0.75c
Next, we calculate the Newtonian momentum:
p_newtonian = m0 * v
Now, we can calculate the percentage error using the following formula:
Percentage error = (p_rel - p_newtonian) / p_rel * 100
Substituting the values we found earlier:
Percentage error = (1.33333 * m0 * 0.75c - m0 * 0.75c) / (1.33333 * m0 * 0.75c) * 100
Simplifying the equation:
Percentage error = (m0 * 0.99999 * 0.75c) / (1.33333 * m0 * 0.75c) * 100
Percentage error = (0.99999 / 1.33333) * 100
Percentage error = 0.749992 * 100
Percentage error ≈ 74.9992%
Therefore, the Newtonian expression for momentum is approximately 74.9992% in error compared to the correct relativistic expression.
To determine the percentage error in the Newtonian expression for the momentum of a particle moving at 0.75c relative to a lab on Earth, we need to compare the classical formula to the correct relativistic formula.
The classical formula for momentum is given by:
p = mv
where p is the momentum, m is the mass, and v is the velocity.
In relativistic mechanics, the correct formula for momentum is:
p = γmv
where γ (gamma) is the Lorentz factor, given by:
γ = 1 / sqrt(1 - (v^2 / c^2))
Here, c is the speed of light in a vacuum.
Now, we can calculate the relative difference between the classical and correct formulas to find the percentage error:
Step 1: Calculate the momentum using the classical formula:
p_classical = mv
Step 2: Calculate the momentum using the relativistic formula:
γ = 1 / sqrt(1 - (0.75c)^2 / c^2)
= 1 / sqrt(1 - 0.5625)
= 1 / sqrt(0.4375)
= 1 / 0.6614
p_relativistic = γmv
= (1 / 0.6614) * mv
Step 3: Calculate the difference between the relativistic and classical momenta:
Δp = p_relativistic - p_classical
Step 4: Calculate the percentage error:
Percentage Error = (Δp / p_relativistic) * 100
By following these steps and substituting the actual values into the calculations, you can determine the percentage error in the Newtonian expression for the momentum of the particle moving at 0.75c relative to the lab on Earth.