Challenging +1.8 This M4 question requires setting up and solving a differential equation for motion with air resistance proportional to v², then integrating to find distance. It involves non-trivial manipulation of the equation of motion (F=ma with resistance mkv² and weight mg), separation of variables, and careful integration with substitution. The specific initial condition and target (speed halving) require precise algebraic manipulation. While systematic, it demands strong calculus skills and multi-step reasoning beyond standard mechanics questions.
2. At time \(t = 0\), a particle \(P\) of mass \(m\) is projected vertically upwards with speed \(\sqrt { \frac { g } { k } }\), where \(k\) is a constant. At time \(t\) the speed of \(P\) is \(v\). The particle \(P\) moves against air resistance whose magnitude is modelled as being \(m k v ^ { 2 }\) when the speed of \(P\) is \(v\). Find, in terms of \(k\), the distance travelled by \(P\) until its speed first becomes half of its initial speed.
2. At time $t = 0$, a particle $P$ of mass $m$ is projected vertically upwards with speed $\sqrt { \frac { g } { k } }$, where $k$ is a constant. At time $t$ the speed of $P$ is $v$. The particle $P$ moves against air resistance whose magnitude is modelled as being $m k v ^ { 2 }$ when the speed of $P$ is $v$. Find, in terms of $k$, the distance travelled by $P$ until its speed first becomes half of its initial speed.
\hfill \mbox{\textit{Edexcel M4 Q2 [8]}}