1) Initial upward acceleration: ![6.0 m/s^2](https://tex.z-dn.net/?f=6.0%20m%2Fs%5E2)
2) Mass of burned fuel: ![0.10\cdot 10^4 kg](https://tex.z-dn.net/?f=0.10%5Ccdot%2010%5E4%20kg)
Explanation:
1)
There are two forces acting on the rocket at the beginning:
- The force of gravity, of magnitude
, in the downward direction, where
is the rocket's mass
is the acceleration of gravity
- The thrust of the motor, T, in the upward direction, of magnitude
![T=3.0\cdot 10^5 N](https://tex.z-dn.net/?f=T%3D3.0%5Ccdot%2010%5E5%20N)
According to Newton's second law of motion, the net force on the rocket must be equal to the product between its mass and its acceleration, so we can write:
(1)
where a is the acceleration of the rocket.
Solving for a, we find the initial acceleration:
![a=\frac{T-mg}{m}=\frac{3.0\cdot 10^5-(1.9\cdot 10^4)(9.8)}{1.9\cdot 10^4}=6.0 m/s^2](https://tex.z-dn.net/?f=a%3D%5Cfrac%7BT-mg%7D%7Bm%7D%3D%5Cfrac%7B3.0%5Ccdot%2010%5E5-%281.9%5Ccdot%2010%5E4%29%289.8%29%7D%7B1.9%5Ccdot%2010%5E4%7D%3D6.0%20m%2Fs%5E2)
2)
When the rocket reaches an altitude of 5000 m, its acceleration has increased to
![a'=6.9 m/s^2](https://tex.z-dn.net/?f=a%27%3D6.9%20m%2Fs%5E2)
The reason for this increase is that the mass of the rocket has decreased, because the rocket has burned some fuel.
We can therefore rewrite eq.(1) as
![T-m'g=m'a'](https://tex.z-dn.net/?f=T-m%27g%3Dm%27a%27)
where
is the new mass of the rocket
Re-arranging the equation and solving for m', we find
![m'=\frac{T}{g+a}=\frac{3.0\cdot 10^5}{9.8+6.9}=1.8\cdot 10^4 kg](https://tex.z-dn.net/?f=m%27%3D%5Cfrac%7BT%7D%7Bg%2Ba%7D%3D%5Cfrac%7B3.0%5Ccdot%2010%5E5%7D%7B9.8%2B6.9%7D%3D1.8%5Ccdot%2010%5E4%20kg)
And since the initial mass of the rocket was
![m=1.9 \cdot 10^4 kg](https://tex.z-dn.net/?f=m%3D1.9%20%5Ccdot%2010%5E4%20kg)
This means that the mass of fuel burned is
![\Delta m = m-m'=1.9\cdot 10^4 - 1.80\cdot 10^4 = 0.10\cdot 10^4 kg](https://tex.z-dn.net/?f=%5CDelta%20m%20%3D%20m-m%27%3D1.9%5Ccdot%2010%5E4%20-%201.80%5Ccdot%2010%5E4%20%3D%200.10%5Ccdot%2010%5E4%20kg)