February 25, 2021

Solve Thursday's Problem

Aerospace engineering is all about designing and building machines that fly. It is comprised of (1) aeronautical engineering, which focuses on aircraft, like jets and helicopters and (2) astronautical engineering, which is all about spacecraft, like satellites, spaceships and even spacesuits. Aerospace engineers must consider multiple factors to build aircraft and spacecraft, such as gravity, speed, fuel consumption, atmospheric pressure and conditions in space, so all types of math are commonly used in this field.

To get a satellite into space, it must be attached to a rocket that can carry it there. Even if a satellite is small, the rocket will require a lot of fuel to lift it...and it will need even more fuel to lift that fuel. Engineers estimate how much fuel will be required with the mass ratio, or minitial /mfinal, where minitial is the initial mass of the rocket (including the rocket itself, the satellite and all its fuel) and mfinal is the final mass of the rocket (after all its fuel has been burned).

A mass ratio of 15, for example, means the rocket will carry 15 times more fuel than the rest of the mass of the rocket. Typical mass ratios range from 8 to 20...in fact, for many launches, fuel makes up the majority of the initial mass!

4.1) To determine the mass ratio for a particular rocket launch, aerospace engineers use the rocket equation:

where change in velocity, the rocket's speed, equals the change in the rocket’s position divided by the change in time and exhaust velocity is the rate at which exhaust gases leave the rocket’s engine.

Coordinate Planes, Inc. launched a satellite, Vector-21, into orbit. The rocket carrying Vector-21 traveled 940,000 meters up in 158 seconds, before releasing the satellite. If the exhaust velocity was 2000 m/s, what is the mass ratio? Express your answer to the nearest tenth.

4.2) Fuel made up 91% of the initial mass (minitial) in the launch of Vector-21. If the final mass (mfinal) was 41,255 kg, how many kilograms of fuel were sent up? Express your answer to the nearest hundredth.

4.3) The Vector-21 satellite travels a total distance of 43,200 km to orbit Earth. Traveling at a rate of 8 km/s, how many minutes does it take the satellite to completely orbit Earth?

4.4) Given that Earth has a circumference of 40,075 km, how many kilometers above Earth’s surface does the Vector-21 orbit? Express your answer to the nearest tenth.