What mass will a spaceship have after firing its engines to reach a new speed?

To find the mass of the spaceship after it fires its engines, we can use the rocket equation, also known as Tsiolkovsky’s rocket equation. This equation relates the change in velocity of a rocket to the effective exhaust velocity and the initial and final masses of the rocket.

The rocket equation is given by:

Δv = v_e * ln(m₀ / m_f)

Where:

• Δv = change in velocity (final velocity – initial velocity)
• v_e = exhaust velocity of the rocket
• m₀ = initial mass of the spaceship
• m_f = final mass of the spaceship after fuel has been burned

Let’s calculate the change in velocity (Δv):

Initial velocity (v_i) = 6.40 x 10³ m/s
Final velocity (v_f) = 7.77 x 10³ m/s

Now, we calculate Δv:

Δv = v_f – v_i = (7.77 x 10³ m/s) – (6.40 x 10³ m/s) = 1.37 x 10³ m/s

Next, we know that the exhaust velocity (v_e) is:

v_e = 4.73 x 10³ m/s

Substituting these values into the rocket equation, we have:

1.37 x 10³ = (4.73 x 10³) * ln(m₀ / m_f)

Now, we need to find the initial mass (m₀) which is given as:

m₀ = 5.10 x 10⁴ kg

Rearranging the equation to solve for ln(m₀ / m_f):

ln(m₀ / m_f) = Δv / v_e = (1.37 x 10³) / (4.73 x 10³)

Calculating this gives:

ln(m₀ / m_f) ≈ 0.2894

To eliminate the natural logarithm, we exponentiate both sides:

m₀ / m_f = e^0.2894 ≈ 1.333

Now we can rearrange this to find the final mass (m_f):

m_f = m₀ / 1.333

Substituting the initial mass:

m_f = (5.10 x 10⁴ kg) / 1.333 ≈ 3.83 x 10⁴ kg

Thus, the mass of the spaceship after it fires its engines to reach a speed of 7.77 x 10³ m/s will be approximately 3.83 x 10⁴ kg.