3. A toy car of mass 0.2 kg is travelling in a straight line on a horizontal floor. The car is modelled as a particle. At time \(t = 0\) the car passes through a fixed point \(O\). After \(t\) seconds the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car is at a point \(P\) with \(O P = x\) metres. The resultant force on the car is modelled as \(\frac { 1 } { 10 } x ( 4 - 3 x ) \mathrm { N }\) in the direction \(O P\). The car comes to instantaneous rest when \(x = 6\). Find

1. an expression for \(v ^ { 2 }\) in terms of \(x\),
2. the initial speed of the car. \section*{4.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{c0a3336d-b0ca-4588-80d1-445e2a5e493c-3_1022_633_268_760}
   A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings \(A P\) and \(B P\) each of length \(l\). The ends \(A\) and \(B\) are attached to fixed points, with \(A\) vertically above \(B\) and \(A B = \frac { 3 } { 2 } l\), as shown in Fig. 1. The particle \(P\) moves in a horizontal circle with constant angular speed \(\omega\). The centre of the circle is the mid-point of \(A B\) and both strings remain taut.
3. Show that the tension AP is \(\frac { 1 } { 6 } m \left( 3 l \omega ^ { 2 } + 4 g \right)\).
4. Find, in terms of \(m , l , \omega\) and \(g\), an expression for the tension in \(B P\).
5. Deduce that \(\omega ^ { 2 } \geq \frac { 4 g } { 3 l }\).