Q11E - University Physics with Modern Physics

Question

An astronaut is inside a $\mathbf{2}\mathbf{.}\mathbf{25}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }\mathbf{kg}$  rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4g.

(a) What is the maximum initial thrust this rocket’s engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket. 

(b) What force, in terms of the astronaut’s weight w, does the rocket exert on her? Start with a free-body diagram of the astronaut. 

(c) What is the shortest time it can take the rocket to reach the speed of sound?

Step-by-Step Solution

Answer

(a) The maximum initial thrust of the engine is $1.10\times {10}^8 \mathrm{N}$ .

(b) The force exerted on astronauts by rocket is 5w .

(c) The shortest time for a rocket to achieve the speed of sound is 8.4 s .

Step 1: Maximum Thrust

Given Data:

• The mass of the rocket, $m=2.25\times {10}^6 \mathrm{kg}$.
• 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.

Step 2: Determine the maximum initial thrust

(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,

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

 

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

Step 3: Determine the force exerted on the astronaut

(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 .

Step 4: Determine the shortest time for the rocket to achieve the speed of sound

(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,

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

 

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