Final answer:
The velocity of the spacecraft with a mass ratio of 20 and an exhaust velocity of 2.7 km/s can be found using Tsiolkovsky's rocket equation, which yields the velocity after multiplying the exhaust velocity by the natural logarithm of the mass ratio.
Step-by-step explanation:
To find the velocity of the spacecraft, we can use the rocket equation, also known as Tsiolkovsky's rocket equation, which relates the varying mass of the rocket to its velocity. The equation is:
v = Isp * g * ln(mr)
Where v is the velocity of the spacecraft, Isp is the specific impulse (which is the exhaust velocity divided by g, the acceleration due to gravity), g is 9.81 m/s² (standard gravity), ln is the natural logarithm, and mr is the mass ratio.
Given the mass ratio (mr) of 20, an exhaust velocity of 2.7 km/s (which is 2,700 m/s), and knowing that g is approximately 9.81 m/s², we can calculate the spacecraft's velocity:
v = 2700 m/s * ln(20)
After calculating the natural logarithm of the mass ratio and multiplying by the exhaust velocity, the final velocity of the spacecraft should be determined.
Keep in mind that the firing time is not used in the rocket equation, as the equation assumes instantaneous velocity change. However, in real-world scenarios, firing time can affect how the velocity change is applied over time.