1) Initial upward acceleration: [tex]6.0 m/s^2[/tex]
2) Mass of burned fuel: [tex]0.10\cdot 10^4 kg[/tex]
Explanation:
1)
There are two forces acting on the rocket at the beginning:
- The force of gravity, of magnitude [tex]F_g = mg[/tex], in the downward direction, where
[tex]m=1.9\cdot 10^4 kg[/tex] is the rocket's mass
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
- The thrust of the motor, T, in the upward direction, of magnitude
[tex]T=3.0\cdot 10^5 N[/tex]
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:
[tex]T-mg=ma[/tex] (1)
where a is the acceleration of the rocket.
Solving for a, we find the initial acceleration:
[tex]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[/tex]
2)
When the rocket reaches an altitude of 5000 m, its acceleration has increased to
[tex]a'=6.9 m/s^2[/tex]
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
[tex]T-m'g=m'a'[/tex]
where
[tex]m'[/tex] is the new mass of the rocket
Re-arranging the equation and solving for m', we find
[tex]m'=\frac{T}{g+a}=\frac{3.0\cdot 10^5}{9.8+6.9}=1.8\cdot 10^4 kg[/tex]
And since the initial mass of the rocket was
[tex]m=1.9 \cdot 10^4 kg[/tex]
This means that the mass of fuel burned is
[tex]\Delta m = m-m'=1.9\cdot 10^4 - 1.80\cdot 10^4 = 0.10\cdot 10^4 kg[/tex]
(a) The initial upward acceleration of the rocket is 6.0 m/s².
(b) The mass of the fuel that burned is 1,000 kg.
The given parameters;
(a) The acceleration of the rocket is calculated as follows;
T = mg + ma
[tex]ma = T- mg\\\\a = \frac{T-mg}{m} \\\\a = \frac{(3\times 10^5) - (1.9\times 10^4 \times 9.8)}{1.9\times 10^4} \\\\a = 6.0 \ m/s^2[/tex]
(b) The mass of the fuel that burned is calculated as follows;
[tex]T = m_2a + _2g\\\\T = m_2 ( a + g)\\\\m_2 = \frac{T}{a+ g} \\\\m_2 = \frac{3\times 10^5}{6.9 + 9.8} \\\\m_2 = 1.8 \times 10^4 \ kg[/tex]
The mass of the fuel = [tex]1.9\times 10^4 - 1.8 \times 10^4 = 1 \times 10^3 \ kg = 1,000 \ kg[/tex]
"Your question is not complete it seems to be missing the following information;"
1.9×104 kg rocket has a rocket motor that generates 3.0×105 N of thrust.
(a) What is the rocket's initial upward acceleration? Express your answer to two significant figures and include the appropriate units
(b) At an altitude of 5000 m the rocket's acceleration has increased to 6.9 m/s2 . What mass of fuel has it burned?
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