Consider a Falcon $9,$ a two$-$state$-$to$-$orbit launch vehicle. The take$-$off mass of Falcon $9$ is $m_{to}=550,000kg,$ the first stage fuel burn rate is $m^{}=2500k\frac{g}{s}$ over a maximum of $160$

s $,$ and constant thrust of $T=7500kN.$ Making a lot of simplifying assumptions including constant gravity, no aerodynamic drag, thrust constant with altitude, we can

approximate the vertical acceleration $($assuming vertical thrust$)$ as:

$a_y(t)=\frac{T-m_{0}g+(m^{})gt}{m_0-(m^{})t}$

After $60$ s $,$ lets assumed that the rocket is turned by $45$ deg so that the equations for acceleration become $($assuming flat Earth, i$.$e$.,$ gravity is still in $y-$direction$)$:

$a_x(t)=\frac{(\frac{T}{\sqrt[\phantom{2}]{2}})}{m_0-(m^{})t}$

$a_y(t)=\frac{(\frac{T}{\sqrt[\phantom{2}]{2}})-m_{0}g+(m^{})gt}{m_0-(m^{})t}$

Calculate the magnitude of velocity $($in $k\frac{m}{s})$ at $t=123sec.$