Answer :
The mass of the spacecraft is approximately [tex]261,313.24\ pounds[/tex] mass.
To find the mass of the spacecraft, we can use Newton's second law of motion:
[tex]\[ F = ma \][/tex]
where:
[tex]- \( F \)[/tex] is the force (thrust) provided by the first stage ([tex]53 kN[/tex]),
[tex]- \( m \)[/tex] is the mass of the spacecraft,
[tex]- \( a \)[/tex] is the acceleration ([tex]18,000 \ mph^2[/tex]).
First, let's convert the thrust from kilo-newtons (kN) to newtons (N), since the unit of acceleration is meters per second squared (m/s²):
[tex]\[ 1 \text{ kN} = 1000 \text{ N} \][/tex]
So, [tex]53 kN[/tex] is equivalent to [tex]\(53 \times 1000 = 53000\) N.[/tex]
Now, let's convert the acceleration from miles per hour squared (mph²) to meters per second squared (m/s²).
[tex]\[ 1 \text{ mph} = \frac{1609.34}{3600} \text{ m/s} \][/tex]
[tex]1 mph^2 = \left(\frac{1609.34}{3600}\right)^2 \text{ m/s²}[/tex]
[tex]1 mph^2 = 0.447 m/s^2}[/tex]
We can rearrange Newton's second law to solve for the mass [tex]\( m \)[/tex]
[tex]\[ m = \frac{F}{a} \][/tex]
[tex]\[ m = \frac{53000 \text{ N}}{0.447 \text{ m/s²}} \][/tex]
[tex]\[ m = 118488 \text{ kg} \][/tex]
To convert kilograms to pounds-mass, we use the conversion factor:
[tex]\[ 1 \text{ kg} = 2.20462 \text{ pounds-mass} \][/tex]
So,
[tex]\[ \text{mass (pounds-mass)} = 118488 \text{ kg} \times 2.20462 \][/tex]
[tex]\[ \text{mass (pounds-mass)} = 261313.24 \text{ pounds-mass} \][/tex]
Use Newton's second law to find the mass in kilograms, and finally convert it to pounds-mass, which is approximately 14.52 pounds-mass.
The question asks us to calculate the mass of the spacecraft given the thrust of the first booster stage and the acceleration of the space shuttle. To find the mass, we use Newton's second law of motion, which states that Force = [tex]mass imes acceleration (F = m imes a).[/tex] However, we first need to convert the given acceleration from miles per hour squared to meters per second squared, and then convert the mass obtained in kilograms to pounds-mass.
First, let's convert the acceleration: 18,000 miles/hour2 is approximately 8,046.72 m/s2 (using the conversion factor 1 mile = 1,609.34 meters and 1 hour = 3600 seconds). Next, we can calculate the mass (in kilograms) by rearranging the formula to m = F / a, which gives us 53,000 N / 8,046.72 m/s2
= approximately 6.59 kilograms. We then convert kilograms to pounds-mass (1 kilogram = 2.20462 pounds), resulting in the mass of the spacecraft being approximately 14.52 pounds-mass.
