Challenging +1.2 This is a Further Maths mechanics question requiring knowledge of radial and transverse acceleration components in circular motion. Students must recall that radial acceleration is v²/r and transverse acceleration is dv/dt, set them equal using the given velocity function, and solve a quadratic equation. While it involves multiple steps and FM content, the approach is relatively standard once the formulas are identified.
1 A particle \(P\) is moving in a circle of radius 1.5 m . At time \(t \mathrm {~s}\) its velocity is \(\left( k - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a positive constant. When \(t = 3\), the magnitudes of the radial and transverse components of the acceleration of \(P\) are equal. Find the possible values of \(k\).
1 A particle $P$ is moving in a circle of radius 1.5 m . At time $t \mathrm {~s}$ its velocity is $\left( k - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a positive constant. When $t = 3$, the magnitudes of the radial and transverse components of the acceleration of $P$ are equal. Find the possible values of $k$.
\hfill \mbox{\textit{CAIE FP2 2012 Q1 [4]}}