Chapter 8, Problem 1ICA

The space shuttle fleet was designed with two booster stages.

1. a. If the first stage provides a thrust of 5.25 mega-newtons [MN] and the space shuttle has a mass of 4,470,000 pound-mass [lb_m], what is the acceleration of the spacecraft in miles per hour squared [mi/h²]?
2. b. If the second stage provides a thrust of 75 kilo-newtons [kN] and the space shuttle has an acceleration of 15,000 miles per hour squared [mi/h²], what is the mass of the spacecraft in units of pounds-mass [lb_m]?

To determine

Calculate the acceleration of the spacecraft in miles per hour squared.

Answer to Problem 1ICA

The acceleration of the spacecraft in miles per hour squared is $\underset{\_}{20,857.19\text{mi}/{\text{h}}^{\text{2}}}$.

Explanation of Solution

Given data:

The thrust provided in the first stage is 5.25 MN, and

Mass of the space shuttle is $4,470,000{\text{lb}}_{\text{m}}$.

Formula used:

Write the expression for force.

$F=ma$ (1)

Here,

$F$ is the force,

$m$ is the mass, and

$a$ is the acceleration.

Calculation:

Consider the conversion factor to convert MN to N.

$1\text{MN}={10}^6\text{N}$

Consider the above conversion factor to convert 5.25 MN to N.

$5.25\text{MN}=5.25\times {10}^6\text{N}$

Consider the conversion factor to convert ${\text{lb}}_{\text{m}}$ to kg.

$1{\text{lb}}_{\text{m}}=0.453592\text{kg}$

Consider the above conversion factor to convert $4,470,000{\text{lb}}_{\text{m}}$ to kg.

$\begin{array}{c}1{\text{lb}}_{\text{m}}=4,470,000\times 0.453592\text{kg}\\ =\text{ 2,027,556}\text{.24}\text{kg}\end{array}$

Rearrange equation (1) to solve for the acceleration ‘a’.

$a=\frac{F}{m}$ (2)

Substitute $5.25\times {10}^6\text{N}$ for F and 2,027,556.24 kg for m in equation (2) to obtain the value of acceleration.

$\begin{array}{c}a=\frac{5.25\times {10}^6\text{N}}{2,027,556.24\text{ kg}}\\ =2.589\frac{\text{m}}{{\text{s}}^{\text{2}}}\\ \cong 2.59\frac{\text{m}}{{\text{s}}^{\text{2}}}\end{array}$

Consider the conversion factor to convert $\frac{\text{m}}{{\text{s}}^{\text{2}}}$ to $\frac{\text{mi}}{{\text{h}}^{\text{2}}}$.

$1\frac{\text{m}}{{\text{s}}^{\text{2}}}=8,052.9707\frac{\text{mi}}{{\text{h}}^{\text{2}}}$

Consider the above conversion factor and convert the value $2.59\frac{\text{m}}{{\text{s}}^{\text{2}}}$ to $\frac{\text{mi}}{{\text{h}}^{\text{2}}}$.

$\begin{array}{c}2.59\frac{\text{m}}{{\text{s}}^{\text{2}}}=2.59\left(1\frac{\text{m}}{{\text{s}}^{\text{2}}}\right)\\ =2.59\times 8,052.9707\frac{\text{mi}}{{\text{h}}^{\text{2}}}\\ =20,857.19\text{mi}/{\text{h}}^{\text{2}}\end{array}$

Conclusion:

Hence, the acceleration of the spacecraft in term of miles per hour squared is $\underset{\_}{20,857.19\text{mi}/{\text{h}}^{\text{2}}}$.

To determine

Calculate the mass of the spacecraft in units of pounds mass.

Answer to Problem 1ICA

The mass of the spacecraft in pounds is $\underset{\_}{88,789.55{\text{lb}}_{\text{m}}}$.

Explanation of Solution

Given data:

The thrust provided in the second stage is 75 kN, and

Acceleration of the space shuttle is $15,000\frac{{\text{m}}_{\text{i}}}{{\text{h}}^{\text{2}}}$.

Calculation:

Consider the conversion factor to convert kN to N.

$1\text{kN}={10}^3\text{N}$

Consider the above conversion factor to convert 75 kN to N.

$75\text{kN}=75\times {10}^3\text{N}$

Consider the conversion factor to convert $\frac{\text{mi}}{{\text{h}}^{\text{2}}}$ to $\frac{\text{m}}{{\text{s}}^{\text{2}}}$.

$1\frac{\text{mi}}{{\text{h}}^{\text{2}}}=0.00012417\frac{\text{m}}{{\text{s}}^{\text{2}}}$

Consider the above conversion factor to convert $15,000\frac{{\text{m}}_{\text{i}}}{{\text{h}}^{\text{2}}}$ to $\frac{\text{m}}{{\text{s}}^{\text{2}}}$.

$\begin{array}{c}15,000\frac{{\text{m}}_{\text{i}}}{{\text{h}}^{\text{2}}}=15,000\left(1\frac{{\text{m}}_{\text{i}}}{{\text{h}}^{\text{2}}}\right)\\ =15,000\times 0.00012417\frac{\text{m}}{{\text{s}}^{\text{2}}}\\ =1.86255\frac{\text{m}}{{\text{s}}^{\text{2}}}\end{array}$

Rearrange equation (1) to solve for m.

$m=\frac{F}{a}$ (3)

Substitute $75\times {10}^3\text{N}$ for F and $1.86255\frac{\text{m}}{{\text{s}}^{\text{2}}}$ for a in equation (3) to obtain the value of mass.

$\begin{array}{c}m=\frac{75\times {10}^3\text{N}}{1.86255\frac{\text{m}}{{\text{s}}^{\text{2}}}}\\ =40,267.37kg\end{array}$

Consider the conversion factor to convert kg to ${\text{lb}}_{\text{m}}$.

$1\text{kg}=2.205{\text{lb}}_{\text{m}}$

Consider the above conversion factor and convert the value 40,267.37 kg to ${\text{lb}}_{\text{m}}$.

$\begin{array}{c}40,267.37\text{ kg}=40,267.37\left(2.205{\text{lb}}_{\text{m}}\right)\\ =88,789.55{\text{lb}}_{\text{m}}\end{array}$

Conclusion:

Hence, the mass of the spacecraft in the units of pounds is $\underset{\_}{88,789.55{\text{lb}}_{\text{m}}}$.