Suppose a spacecraft of rest mass 34000 kg is accelerated to 0.206c. How much kinetic energy would it have?
If you used the classical formula for KE, by what percentage would you be in error? Do not enter units.
The exact relativistic KE is
mc^2*[1/sqrt[(1 -(v/c)^2] - 1]
= 3.4*10^5*(3.00*10^8)^2*(1.02192-1)
= 6.71*10^20 J
The nonrelativistic (classical) KE is
(1/2)M V^2 = (1/2)*(3.4*10^5)(6.18*10^7)^2 = 6.49*10^20
The percent error (relative to the correct value) is -3.1%
To calculate the kinetic energy (KE) of the spacecraft, we can use the relativistic formula:
KE = (γ - 1) * mass * c^2,
where γ is the Lorentz factor and c is the speed of light.
To find γ, we can use the formula:
γ = 1 / sqrt(1 - (v/c)^2),
where v is the velocity of the spacecraft and c is the speed of light.
Given:
Mass of spacecraft (m) = 34000 kg
Velocity of spacecraft (v) = 0.206c
Speed of light (c) = 3 x 10^8 m/s
Now we can calculate γ:
v/c = 0.206
(0.206)^2 = 0.042436
1 - 0.042436 = 0.957564
sqrt(0.957564) = 0.978555
γ ≈ 1 / 0.978555 = 1.022051
Now we can calculate the kinetic energy (KE):
KE = (γ - 1) * mass * c^2
KE ≈ (1.022051 - 1) * 34000 kg * (3 x 10^8 m/s)^2
KE ≈ 0.022051 * 34000 kg * (3 x 10^8 m/s)^2
KE ≈ 2.45848 x 10^17 J
By substituting the values into the classical formula:
KE = 0.5 * mass * v^2
KE ≈ 0.5 * 34000 kg * (0.206c)^2
KE ≈ 0.5 * 34000 kg * (0.206 * 3 x 10^8 m/s)^2
KE ≈ 3.74 x 10^15 J
To find the percentage error, we can use the formula:
Error percentage = (|Relativistic KE - Classical KE| / Relativistic KE) * 100
Error percentage ≈ (|2.45848 x 10^17 J - 3.74 x 10^15 J| / 2.45848 x 10^17 J) * 100
Error percentage ≈ 1.52174%
So, if we use the classical formula, we would be in error by approximately 1.52174%.
To calculate the kinetic energy (KE) of a relativistic object, we can use the equation:
KE = (γ - 1) * mc^2
where γ is the Lorentz factor, m is the rest mass of the object, and c is the speed of light.
First, let's determine the Lorentz factor (γ):
γ = 1 / sqrt(1 - (v/c)^2)
where v is the velocity of the object and c is the speed of light.
Given:
Rest mass (m) = 34000 kg
Velocity (v) = 0.206c
We need to convert the velocity into a fraction of the speed of light:
v = 0.206 * c
Now we can substitute the values into the formula to calculate γ:
γ = 1 / sqrt(1 - (0.206c / c)^2)
= 1 / sqrt(1 - 0.042436)
= 1 / sqrt(0.957564)
≈ 1 / 0.9784
≈ 1.0227
Next, we can calculate the kinetic energy using the formula:
KE = (γ - 1) * mc^2
KE = (1.0227 - 1) * (34000 kg) * (c^2)
Now, the speed of light (c) is approximately 3 x 10^8 m/s. We can substitute this value into the equation:
KE = (1.0227 - 1) * (34000 kg) * ((3 x 10^8 m/s)^2)
Calculating this will give us the kinetic energy.