Help Please!
A spaceship with fuel is initially at rest at point CG. Fuel is ejected at the back to propel the payload to the forward direction. After a long time, the total fuel ejected out has a mass of 106 pounds, and center of gravity moving backward at 20 m/s. The forward speed of the payload relative to CG is 150 m/s. The payload mass is _____ pounds.
To determine the payload mass, we first need to understand the principle of conservation of momentum. This principle states that the total momentum in a system is conserved if there are no external forces acting on it.
In this case, the spaceship and the ejected fuel make up our system. Initially, both the spaceship and the fuel are at rest, so their total momentum is zero. After ejecting fuel, the spaceship and the remaining fuel move in opposite directions, but the total momentum of the system should still be zero if no external forces act on it.
Let's break down the given information:
- The total mass of the fuel ejected is 106 pounds.
- The center of gravity (CG) of the spaceship-fuel system moves backward at a velocity of 20 m/s.
- The forward speed of the payload relative to CG is 150 m/s.
Now, let's use the principle of conservation of momentum to solve for the payload mass.
Total initial momentum = Total final momentum
Initially: (Mass of spaceship + Mass of fuel) * 0 = 0
Finally: (Mass of spaceship) * (-20 m/s) + (Mass of fuel) * (150 m/s)
Since the total mass of the fuel ejected is given (106 pounds), we can substitute it into the equation:
(Mass of spaceship) * (-20 m/s) + 106 pounds * (150 m/s) = 0
Simplify the equation:
(Mass of spaceship) * (-20) + 15900 = 0
Solving for the mass of the spaceship:
(Mass of spaceship) * (-20) = -15900
Mass of spaceship = -15900 / (-20)
Mass of spaceship = 795 pounds
Now that we have the mass of the spaceship, we can find the mass of the payload by subtracting the mass of the fuel ejected:
Mass of payload = Mass of spaceship - Mass of fuel ejected
Mass of payload = 795 pounds - 106 pounds
Mass of payload = 689 pounds
Therefore, the mass of the payload is 689 pounds.