AS June 2022 Q3
3. A plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \dfrac{3}{4}\)
A particle \(P\) is held at rest at a point \(A\) on the plane.
The particle \(P\) is then projected with speed \(25\ \text{m s}^{-1}\) from \(A\), up a line of greatest slope of the plane.
In an initial model, the plane is modelled as being smooth and air resistance is modelled as being negligible.
Using this model and the principle of conservation of mechanical energy,
In a refined model, the plane is now modelled as being rough, with the coefficient of friction between \(P\) and the plane being \(\dfrac{3}{5}\)
Air resistance is still modelled as being negligible.
Using this refined model and the work-energy principle,
| Scheme | Marks | AO |
|---|---|---|
| \(mg \times \dfrac{25}{6}\sin\alpha\) | B1 | 1.1b |
| Use of the principle of conservation of mechanical energy | M1 | 3.4 |
| \(\dfrac{1}{2}m \times 25^2 - \dfrac{1}{2}mv^2 = mg \times \dfrac{25}{6}\sin\alpha\) | A1 | 1.1b |
| \(v = 24\ (\text{m s}^{-1})\) (23.99895831… = 24 to 2SF if \(g = 9.81\)) | A1 | 1.1b |
| (4) |
Notes
N.B. If consistent use of a specific value of \(m\), allow all the marks but deduct the final A mark in each part but allow full marks if \(m\)’s have been cancelled or don’t appear.
B1: Seen anywhere
M1: Correct no. of terms, dimensionally correct, condone sign errors and sin/cos confusion
M0 for non-energy methods.
Allow max M1A0A0 if 25/6 not resolved or not resolved correctly in PE term
A1: Correct equation in \(m\), \(g\), \(v\) and \(\alpha\)
A1: cao
| Scheme | Marks | AO |
|---|---|---|
| Resolve perpendicular to the plane | M1 | 3.1a |
| \(R = mg\cos\alpha\) | A1 | 1.1b |
| \(F = \dfrac{3}{5}R\) | B1 | 3.4 |
| WD against friction \(= F \times \dfrac{25}{6}\) | B1 | 3.4 |
| Use of work-energy principle | M1 | 3.1a |
| \(\dfrac{1}{2}m \times 25^2 - \dfrac{1}{2}mv^2 = mg \times \dfrac{25}{6}\sin\alpha + \dfrac{3}{5} \times mg\cos\alpha \times \dfrac{25}{6}\) | A1 A1 | 1.1b 1.1b |
| \(v = 23.2\) or 23 \((\text{m s}^{-1})\) (23.16700… = 23.2 or 23 to 3SF or 2SF if \(g = 9.81\)) | A1 | 1.1b |
| (8) | ||
| (12 marks) |
Notes
N.B. If consistent use of a specific value of \(m\), allow all the marks but deduct the final A mark in each part but allow full marks if \(m\)’s have been cancelled or don’t appear.
M1: Correct no. of terms, dimensionally correct, condone sign errors and sin/cos confusion
A1: Correct equation
B1: Seen anywhere
B1: Seen anywhere
M1: Correct no. of terms, dimensionally correct, condone sign errors and sin/cos confusion
M0 for non work-energy methods
Allow max M1A1A0A0 if 25/6 not resolved or not resolved correctly in PE term
A1: Equation in \(m\), \(g\), \(v\) and \(\alpha\) with at most one error
N.B. If KE terms reversed, only penalise ONCE.
A1: Correct equation in \(m\), \(g\), \(v\) and \(\alpha\)
A1: cao