Question

The energy to accelerate a starship comes from combining matter and antimatter. When this is done the total rest energy of the matter and antimatter is converted to other forms of energy. Suppose a starship with a mass of \(2.0 \times 10^{5} \mathrm{kg}\) accelerates to \(0.3500 c\) from rest. How much matter and antimatter must be converted to kinetic energy for this to occur?

Short answer

Answer: To calculate the amount of matter and antimatter required, follow these steps: 1. Calculate the Lorentz factor, γ, using the formula: γ = 1 / sqrt(1 - (v^2 / c^2)). For a speed of 0.3500c, γ is approximately 1.1843. 2. Calculate the final kinetic energy, K, using the formula: K = (γ - 1) * mc^2. With the mass of the starship given as 2.0 x 10^5 kg, K is approximately 4.667 x 10^18 Joules. 3. Find the amount of matter and antimatter, Δm, using the equation: Δm = K / c^2. The required amount of matter and antimatter is approximately 5.185 x 10^10 kg.

Step-by-step solution

Find the final kinetic energy of the starship

First, we need to calculate the final kinetic energy of the starship using the relativistic kinetic energy formula, which is given by: \(K = ( \gamma - 1)mc^2\) where \(K\) is the kinetic energy, \(\gamma\) is the Lorentz factor, \(m\) is the mass of the object, and \(c\) is the speed of light. The Lorentz factor is given by: \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\) Given that \(m = 2.0 \times 10^5 \mathrm{kg},\ v = 0.3500c\). We will first calculate the Lorentz factor and then find the kinetic energy.

Calculate the Lorentz factor

We have the formula for the Lorentz factor: \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\) Substituting the values, we get: \(\gamma = \frac{1}{\sqrt{1-(0.3500)^2}}\) Solve for \(\gamma\).

Calculate the kinetic energy

Now that we have the Lorentz factor, we can find the kinetic energy using the relativistic kinetic energy formula: \(K = ( \gamma - 1)mc^2\) Substitute the values: \(K = (\gamma - 1)(2.0 \times 10^5 \mathrm{kg})(3.0 \times 10^8 \mathrm{m/s})^2\) Solve for \(K\).

Find the amount of matter and antimatter

The total amount of matter and antimatter converted is equal to the total kinetic energy of the starship. We can find this using the rest energy equation: \(E = mc^2\) We already know the value of \(K\) and \(c\). Let's implement the equation like this: \(K = (\Delta m) c^2\) Now we need to solve for \(\Delta m\): \(\Delta m = \frac{K}{c^2}\) Substitute the values of \(K\) and \(c\) to find the required amount of matter and antimatter.