91Ó°ÊÓ

Given Data:

• The mass of the rocket,$m=2.25×{10}^6\mathrm{kg}$.
• The acceleration for blacking out,$a=4g$.

Maximum Thrust:

The maximum thrust that a rocket can provide is equal to the sum of the rocket's weight and force generated by the rocket, with an acceleration equal to blacking out.

(a)

The maximum initial thrust can be calculated as:

$$\begin{gathered} T-mg=ma \\ T-mg=m\left(4g\right) \\ T=5mg \end{gathered}$$

Here, g is the gravitational acceleration, and its value is $9.8\mathrm{m}/{\mathrm{s}}^2$, a is the acceleration for blackout condition, T is the maximum initial thrust.

Substitute all the values in the above equation, and we get,

$$\begin{gathered} T=5\left(2.25×{10}^6\mathrm{kg}\right)\left(9.8\mathrm{m}/{\mathrm{s}}^2\right) \\ T=1.10×{10}^8\mathrm{N} \end{gathered}$$

Therefore, the maximum initial thrust of the engine is $1.10×{10}^8\mathrm{N}$.

(b)

The force of the rocket in terms of the weight of the astronaut can be calculated as:

$$\begin{gathered} N-w=ma \\ N-w=\left(\frac{w}{g}\right)\left(4g\right) \\ N=w+4w \\ N=5W \end{gathered}$$

Here, w is the weight of the astronaut and N is the force of the rocket.

Therefore, the force exerted on astronauts by rockets is 5w .

(c)

The shortest time for a rocket to achieve the speed of sound is calculated as:

$v=u+at$

Here, $v$ is the speed of sound and $u$ is the initial speed of the rocket.

Substitute all the values in the above equation, and we get,

role="math" localid="1663739255474" $$\begin{gathered} 331m/s=0+\left(4g\right)t \\ t=\frac{331\mathrm{m}/\mathrm{s}}{4\left(9.8\mathrm{m}/{\mathrm{s}}^2\right)} \\ t=8.4\mathrm{s} \end{gathered}$$

Therefore, the shortest time for a rocket to achieve the speed of sound is $8.4\mathrm{s}$.