AS June 2018 Q2
2. A car moves round a bend which is banked at a constant angle of \(\theta^\circ\) to the horizontal.
When the car is travelling at a constant speed of \(80\ \text{km h}^{-1}\) there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius \(500\ \text{m}\).
The speed of the car is increased so that it is now travelling at a constant speed of \(90\ \text{km h}^{-1}\)
The car is still modelled as a particle moving in a horizontal circle of radius \(500\ \text{m}\).
| Scheme | Marks | AO |
|---|---|---|
| Complete strategy to find value of \(\theta\) | M1 | 3.1b |
 | ||
| Resolve vertically | M1 | 3.1b |
| \(R\cos\theta^\circ = mg\) | A1 | 1.1b |
| Resolve horizontally | M1 | 3.1b |
| \(R\sin\theta^\circ = \dfrac{mv^2}{r}\) | A1 | 1.1b |
| \(v = 80\ \text{km h}^{-1} = \dfrac{80 \times 1000}{60^2}\ \text{m s}^{-1}\) | B1 | 1.2 |
| Solve simultaneous equations and substitute \(v\) in correct units to obtain \(\theta\): \(\tan\theta^\circ = \dfrac{v^2}{rg} = \dfrac{640000}{36^2 \times 500 \times 9.8}\), \(\theta = 5.8\) | A1 | 2.2a |
| (7) |
Notes
M1: Complete strategy involving resolving in perpendicular directions, change of units and solution of simultaneous equations
M1: Complete strategy to form one equation involving \(\theta\) e.g. resolve vertically. Condone sin/cos confusion
A1: Or equivalent
M1: Complete strategy to form a second equation involving \(\theta\) e.g. resolve horizontally. Condone sin/cos confusion
A1: Correct unsimplified – need not substitute for \(v\) or \(r\)
B1: Correct conversion \(\text{km h}^{-1}\) to \(\text{m s}^{-1}\) (22.2)
A1: Accept 5.8 or 5.75 (follows use of 9.8)
| Scheme | Marks | AO |
|---|---|---|
| All weight acting at a single point | B1 | 3.5b |
| (1) |
Notes
B1: Any appropriate comment
e.g. Only one point of contact with the road
The centre of mass of the car is on the road.
| Scheme | Marks | AO |
|---|---|---|
| Friction acting down the slope | B1 | 2.2a |
| (1) | ||
| (9 marks) |
Notes
B1: Need to include the direction