Question

1.) You are in a spaceship at rest in deep space, outside the influence of any gravity. Without fuel, your vessel has a mass of 5,750 kg and its rocket engine expels exhaust at a speed of 8,250 m/s. You want to travel to an outpost that is quite far away, and you decide to accelerate to a speed of 12,500 m/s, coast for some time, and then slow down to a stop. A) What is the minimum mass of fuel you must have at the beginning of the journey, assuming there is no opportunity to refuel, and that gravity and other external forces are negligible? B) If the engine burns the fuel at a constant rate of 4.75 kg/s, what is the initial acceleration of the spaceship?

Answer 1

Since the space ship is in deep space all the other forces can be neglecte here, and initial velocity is zero. Final mass ( After exhausting all the fuel ) is given.

Using the Tsiolkovsky formula,

The equation relates the delta-v (the maximum change of velocity of the rocket if no other external forces act, which is just the ideal case here) to the effective exhaust velocity and the initial and final mass of a rocket.

${\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}}}$ .......................(i)

In this case, the first part of the equation is enough.

where:

• m₀ is the initial total mass, including propellant,
• m₁ is the final total mass,
• v_e is the effective exhaust velocity,
• Δv is delta-v - the maximum change of velocity of the vehicle (with no external forces acting),
• ln refers to the natural logarithm function.

delta-V in this case is equal to 12500 m/s, since intially the space ship is at rest. v = 0.

v_e = 8250 m/s ( Given )

final mass ( m₁)= 5750 kg.

(a) Putting all these values in equation (i)

mo 12500 = 8250 In 5750

mo In 5750 = 1.51

mo 5750 = exp 1.51

m_{1 = 26,028 kg}

Mass of the fuel = 26028 - 5750 = 20,278 kg (Ans)

(b) Thrust(Force) can be calculated as -

......................(ii)

Am T =VE A

Again, v_{e = 8250 m/s}

_and,

ܐܐܠ rg / s /475=ܨ

so, equatio (ii) becomes

T = 8250 x 4.75 N = 39187.5 N

now, Thrust can be equated to ma, where m is the total initial mass which we found to be 26028 kg

so, 26028 x a = 39187.5

a = 39187.5/26028 m/s²

a = 1.505 m/s². (Ans)