= Solution
{c}
Take upward <velocity> as positive, and let $w>0$ be the rearward <rocket exhaust speed>. During $dt$, the ejected parcel has <mass> $-dm>0$ and laboratory <velocity> $v-w$ to first order. The <momentum> change of the rocket and this parcel equals the gravitational <impulse>:
$$
(m+dm)(v+dv)+(-dm)(v-w)-mv=-mg\,dt.
$$
Discarding the second-order product $dm\,dv$ gives the vertical <rocket equation>
$$
\boxed{m\frac{dv}{dt}=-w\frac{dm}{dt}-mg.}
$$
With $\dot m=-\alpha$ and $m=m_0-\alpha t$, integration for free flight starting from rest at $t=0$ gives
$$
\boxed{v(t)=w\ln\frac{m_0}{m_0-\alpha t}-gt,\qquad 0\le t\le T.}
$$
The logarithmic term is the ideal <Tsiolkovsky rocket equation> contribution; $gt$ is the gravity loss. Immediate upward departure requires positive initial <acceleration>, or
$$
\boxed{\alpha w>m_0g,\qquad w_{\mathrm{threshold}}=\frac{m_0g}{\alpha}.}
$$
At exact equality the initial <acceleration> is zero, but further burning reduces the weight and permits departure immediately thereafter. If departure is allowed to be delayed while fuel burns on the pad, the condition is instead $\alpha w>m(t)g$ at some time before burnout. With final <rocket dry mass> $m_f$, any powered departure before burnout requires $w>gm_f/\alpha$; the free-flight formula then starts at the actual departure time $t_\ell$ with $m(t_\ell)$, giving $v=w\ln[m(t_\ell)/m(t)]-g(t-t_\ell)$.
For the specified data, the initial <mass> includes the payload. The <rocket fuel fraction> gives
$$
m_f=(1-f)m_0=2500\,\mathrm{kg},\qquad
T=\frac{fm_0}{\alpha}=15\,\mathrm s.
$$
The $100\,\mathrm{kg}$ payload is already part of the $2500\,\mathrm{kg}$ final <mass>; it is not the entire <rocket dry mass> and must not be added a second time. Here $m_0g/\alpha=196\,\mathrm{m/s}$, well below the stated <rocket exhaust speed>, so departure is immediate. The burnout <speed> is
$$
\boxed{v(T)=2500\ln4-9.8(15)\simeq3319\,\mathrm{m/s}.}
$$
This is the <speed> at burnout. Subsequent motion under the assumed uniform <gravity> has $v(t)=v(T)-g(t-T)$.
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