7 It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~ms} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-5_414_624_356_242} In a model, the following modelling assumptions are made.

• The track is smooth.
• The car is a particle.
• The car follows a horizontal circular path with radius \(r \mathrm {~m}\).
  1. Show that, according to the model, \(\sqrt { 3 } \mathrm { v } ^ { 2 } = \mathrm { gr }\).

For a particular portion of banked track, \(r = 24\).

Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.

Explain

• how this contrasts with what the model predicts,
• how to improve the model to account for this.