$1)$ Two$-$Stage Launch Vehicle Performance

SpaceX's Falcon $9$ launch vehicle is a multi$-$stage rocket with two stages. The first stage uses nine Merlin $1$D $($LOX$/$Kerosene$)$ engines. The second stage uses a single Merlin $1$D Vacuum $($which has a larger nozzle for superior coefficient of thrust, and thus Isp, at high atmospheric conditions$).$ The stage characteristics are provided in the following table. Write by hand the order of equations you will use to solve the following. Please use Excel for your solutions.

Notes: TLI is Trans Lunar Injection. Also, as discussed in class, the launch community often uses imperial units.

a$.$ Calculate the burn time associated with each stage: Main Engine Cut$-$Off $($MECO$),$ and Second Engine Cut$-$Off $($SECO$)$

b$.$ Estimate the payload capability $\backslash (\backslash$left$(\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$PL$\}\}\backslash$right$)\backslash )$ of the vehicle to LEO $\backslash (\backslash$left$(\backslash$Delta V$\_\{\backslash$text $\{$LEO $\}\}=32,800\backslash$mathrm$\{$ft$\}/\backslash$mathrm$\{$s$\}\backslash$right$)\backslash ).$

c$.$ Compare your calculated payload capabilities to LEO to those you can find online. If they disagree, explain why.

d$.$ Calculate the total impulse associated with each stage. Use these values and the propellant weight to estimate the Isp of the engines. Compare you values to the Isp values in the above table.