Stopping on a dime satellite.

[kosherkittens]

No source of friction in space to slow you down. So if you want to stop motion, you have to rely on other forces. Consider the problem of a moving satellite using only the following commands:

1) turn on the main thruster

2) tun off the main thruster

3) rotate 180 degrees

how fast can you make the satellite traverse a linear distance of exactly 1500 km (starting and stopping with a velocity of 0)? When answering , yous hould consider the following information:

-fully fueled, the satellite's mass is 2420 kg.

-the main thruster produces a constant force of F=96.8 N and consumes fuel at a rate of 1.6 grams/second.

-The rotational thrusters consume .8 g/s

-the satellite requires 400 seconds to complete a rotation of 180 degrees

- cannot use F=ma because in this scenario the satellites thrusters accelerate it by ejecting mass.Instead you should use a more general F= dp/dt, where p denotes momentum ( the product of mass and velocity, which depend on time)

-because the mass of the satellite at the beginning of the trip is not the same as the mass at toward the end, the duration of the first and last legs of this 1500 km trek will not be the same.

 

[gneill]

Satellites are objects that orbit another body, typically in circular or elliptical orbits. It's going to take some fancy work with the thrusters to perform a linear trajectory that starts and ends with zero velocity!

 

[Filip Larsen]

Please show us what equations you think are relevant to solving this problem, and try explain what you have already tried and where you are stuck. If you haven't been able to start solving this problem at all, I would suggest that you try break up the maneuver into segments such that you can attach some equations to each segment that when taken together will allow you to calculate the answer. You will most likely need to make assumptions along the way.

For instance, in the rotation part you know the satellite takes 400 seconds to rotate 180 degrees and it uses 0.8 g/s. Assuming the rotational truster are on for all 400 seconds (more or less half the time to increase rotational speed and the remaining time to decrease it back down to zero) you can write up equations that model how much fuel is used in that segment and, assuming constant speed along the 1500 m "track" during the rotational segment, how much distance it has covered in those 400 seconds.

 

[HallsofIvy]

If m is variable, then you are correct that "F= ma" cannot be used. However, the more general formula
[tex]F= \frac{dmv}{dt}= m\frac{dv}{dt}+ v\frac{dm}{dt}[/tex]
can be used.

 

[rwisz]

I would approach this problem by using the conservation of momentum principle. Since there is no friction in space, the only force acting on the satellite will be the thrust from the main and rotational thrusters. Therefore, we can use the equation F=dp/dt, where F is the thrust force, p is the momentum, and t is time.

First, we need to determine the initial and final momentum of the satellite. At the beginning of the trip, the satellite has an initial velocity of 0 and a mass of 2420 kg. Therefore, the initial momentum is 0 kg*m/s. At the end of the trip, the satellite must also have a velocity of 0, so the final momentum will also be 0 kg*m/s.

Next, we can calculate the change in momentum required to stop the satellite. This will be equal to the initial momentum, or 0 kg*m/s. Since we know the mass of the satellite and the time it takes to rotate 180 degrees, we can calculate the change in velocity required for the satellite to stop. Using the equation p=mv, we can rearrange it to solve for v, which gives us a change in velocity of 1.21 m/s.

Now, we can use the equation F=dp/dt to calculate the thrust force required to achieve this change in momentum. Using the given information, we can calculate that the total mass of fuel consumed during this trip is 640 grams (1.6 g/s * 400 s). Therefore, the total change in momentum will be equal to the mass of the satellite plus the mass of fuel consumed, multiplied by the change in velocity. This gives us a thrust force of 774.4 N.

Next, we need to determine the duration of the first and last legs of the trip. Since the mass of the satellite is changing due to the consumption of fuel, the duration of these legs will also be different. Using the equation F=dp/dt, we can calculate the duration of the first and last legs to be 1071.4 seconds.

Now, we can calculate the duration of the middle leg of the trip, where the satellite is rotating 180 degrees. Since the rotational thrusters consume 0.8 g/s of fuel, the total mass of fuel consumed during this rotation will be 320 grams. Using the same equation, we can calculate the duration of this leg to be 400 seconds.