Okay, So i'm not really into physics or mathematics but i'm trying my best to learn some bit of how the calculations needed for the amount of antimatter needed to propel a spacecrafft to relativistic speeds.
So, according to the wiki, The ISV class of ships is accelerated to 0.7c for 6 to 7 months straight from Earth to Alpha-Centauri and vice versa.
It carries approximately 250 metric tons of cargo at full capacity which included all 250 passengers, two valkyrie shuttles and the cargo racks loaded.
Let's recalculate the amount of antimatter required for a longer duration of constant acceleration, specifically 6.5 months, which is approximately 4,745 hours:
1. Spacecraft Mass: Assume a spacecraft mass (including cargo, colonists, shuttles, and structure) of 250 metric tons, which is approximately 250,000 kilograms.
2. Desired Acceleration: Let's assume the spacecraft accelerates at a constant rate for a 6.5 Months (4,745 Hours). The desired final velocity is 0.7c (70% of the speed of light).
3. Relativistic Effects: When approaching the speed of light, relativistic effects become significant. To account for this, we'll use the relativistic rocket equation:
Δv = c * tanh⁻¹(a * Δt / c)
Δv is the change in velocity (0.7c - 0c = 0.7c).
a is the constant acceleration.
Δt is the time duration in the spaceship's frame (4,745 hours).
4. Antimatter Propulsion Efficiency: We'll assume 100% conversion efficiency of antimatter to energy (which is an idealized assumption and may not be achievable in practice).
Now, we can solve for the required acceleration (a):
Now, we calculate the required energy (E) to achieve this acceleration:
E = (1/2) * m * Δv²
Where m is the spacecraft mass and Δv is the change in velocity.
E ≈ (1/2) * 250,000 kg * (0.7c)²
Next, we convert the energy to antimatter mass using Einstein's mass-energy equivalence (E = mc²):
m = E / c²
Plugging in the values:
m ≈ [(1/2) * 250,000 kg * (0.7c)²] / c²
m ≈ 0.000214 kilograms of antimatter
So, approximately 0.000214 kilograms (or 0.214 grams) of antimatter would be required to propel the spacecraft to 0.7 times the speed of light (0.7c) over a duration of 6.5 months with the corrected assumptions.
Is my math correct in this?
And if that's the case... THAT'S ALL THE AMOUNT OF ANTIMATTER NEEDED to propel such a big ass ship??? not even 1 kg? Is this how powerful antimatter is?
I mean of course producing it is the riskiest economic suicide ever but I never thought antimatter would be absolutely this intense.
Edit: I made a mistake. I literally calculated 0.7 the decimal as opposed to 0.7 light speed (209854720.6) and only used c as opposed to c². its indeed 61,250 kgs of antimatter for the 250,000.