AS June 2025 Q4

4. A car is travelling round a circular track. The car moves with constant speed in a horizontal circle of radius \(r\).

In an initial model,

• the car and driver are modelled as a single particle
• the track is modelled as being rough, so that there is sideways friction between the tyres of the car and the track with coefficient of friction \(\mu\)
• the track is modelled as being horizontal

Using this model, the maximum speed at which the car can move round the circle of radius \(r\) without slipping sideways is \(\dfrac{1}{2}\sqrt{gr}\).

In a refined model,

• the car and driver are modelled as a single particle
• the track is modelled as being rough, so that there is sideways friction between the tyres of the car and the track with coefficient of friction \(\dfrac{1}{4}\)
• the track is modelled as being banked at an angle \(\theta\) to the horizontal

Using this model, the minimum speed at which the car can move round the circle of radius \(r\) without slipping sideways is \(\sqrt{\dfrac{4rg}{35}}\)