Finding relative speed of space shuttle

[huybinhs]

Homework Statement

The space shuttle, with an initial mass M = 2.81E+6 kg, is launched from the surface of the Earth with an initial net acceleration a = 17.5 m/s2. The rate of fuel consumption is R = 8.20E+3 kg/s. The mass of the shuttle when it reaches orbit is Mo = 6.74E+5 kg. What is the relative speed of the exhaust gas?

The Attempt at a Solution

I got exhaust v = ma dt /dm = 6 km/s

then I got relative speed = exhaust v * (M/Mfuel)

which Mfuel = (2.81* 10^6) - (6.74*10^5) = 2.136 * 10^6 kg

=> I got the relative speed = 1.65 kg/s which is NOT CORRECT!

Please help!

 

[Mindscrape]

Your equation for the speed isn't right.

Vshuttle_final=Vshuttle_initial+v_exhaust*ln(Mf/Mi)+a*t

Where I assume that the initial and final velocities of the shuttle are zero, starts from rest, and ends at orbit (rest).

Also, the fuel consumption will looks like this
dm/dt=R
mf=mi+Rt
which allows you to solve for t

I don't know what level of physics this is, so if you don't know differential equations (I assume you don't as this is "Introductory Physics") then I guess you have some formulas that you never listed. If you have done differential equations, work it out for yourself to see if what I say is true.

 

[huybinhs]

Ok, thanks for your help! ;)

 

[Mindscrape]

Actually, I don't really know what I was saying earlier (it was the morning), you ought to know how high the shuttle orbits because you need that velocity as your final velocity, but the problem doesn't give it, huh...

 

[rwisz]

I would approach this problem by first analyzing the given information and identifying the relevant equations and variables. In this case, we have the initial mass of the space shuttle (M), the initial acceleration (a), the rate of fuel consumption (R), and the final mass of the shuttle when it reaches orbit (Mo). We also need to consider the mass of the exhaust gas (Mfuel) and the relative speed of the exhaust gas (v).

To find the relative speed of the exhaust gas, we can use the conservation of momentum equation:

Mv = Mfuel * v_exhaust

where v_exhaust is the speed of the exhaust gas.

We can also use the conservation of mass equation:

M - Mfuel = Mo

Combining these equations, we can solve for the relative speed of the exhaust gas:

v_exhaust = (M/Mfuel) * (M - Mo)

Substituting the given values, we get:

v_exhaust = (2.81E+6 kg / 2.136E+6 kg) * (2.81E+6 kg - 6.74E+5 kg)

= 1.65E+3 m/s

Therefore, the relative speed of the exhaust gas is approximately 1.65 km/s. It is important to note that this is an approximate value as we have made some assumptions and have not accounted for factors such as air resistance and changes in gravitational pull during the launch. To obtain a more precise value, we would need to consider these factors and perform more detailed calculations.