The space shuttle, with an initial mass M = 2.41×106 kg, is launched from the surface of the earth with an initial net acceleration a = 26.1 m/s^2. The rate of fuel consumption is R = 6.90×103 kg/s. The shuttle reaches outer space with a velocity of vo = 4632. m/s, and a mass of Mo = 1.45×106 kg. How much fuel must be burned after this time to reach a velocity vf = 5285. m/s?

should I apply force of gravity into this question?? or concept of impluse

To solve this question, you will need to apply the concept of impulse. Impulse is defined as the change in momentum of an object, and it is equal to the force exerted on the object multiplied by the time interval over which the force acts.

In this case, you can calculate the initial momentum of the space shuttle using the formula:

initial momentum = mass x initial velocity

Then, you need to apply the concept of impulse to determine the change in momentum required to reach the final velocity. The change in momentum is given by:

change in momentum = final momentum - initial momentum

Since the mass of the space shuttle changes over time due to fuel consumption, you will need to consider the rate of fuel consumption. The rate of fuel consumption is given by:

rate of fuel consumption = mass lost / time

You are given the initial mass of the shuttle (M) and the mass of the shuttle after reaching outer space (Mo). From this, you can calculate the mass lost:

mass lost = M - Mo

To calculate the time required to burn this amount of fuel, you can divide the mass lost by the rate of fuel consumption:

time = mass lost / rate of fuel consumption

Once you have the time, you can calculate the force exerted on the shuttle using the formula:

force = mass x acceleration

Substituting the values for mass and acceleration, you can calculate the force.

Finally, you can apply the concept of impulse to determine the change in momentum required to reach the final velocity. The impulse is given by:

impulse = force x time

Using this impulse, you can calculate the change in momentum. Then, you can add this change in momentum to the initial momentum to obtain the final momentum.

Finally, you can calculate the amount of fuel required to reach the final velocity by rearranging the initial momentum formula:

fuel required = (final momentum - initial momentum) / rate of fuel consumption

Substituting the values, you can calculate the fuel required to reach the final velocity vf.