October 2021 Paper 3 Mechanics Q4
4.
A small stone is projected with speed \(65\ \text{m s}^{-1}\) from a point \(O\) at the top of a vertical cliff.
Point \(O\) is 70 m vertically above the point \(N\).
Point \(N\) is on horizontal ground.
The stone is projected at an angle \(\alpha\) above the horizontal, where \(\tan\alpha = \dfrac{5}{12}\)
The stone hits the ground at the point \(A\), as shown in Figure 3.
The stone is modelled as a particle moving freely under gravity.
The acceleration due to gravity is modelled as having magnitude \(\textbf{10 m s}^{\mathbf{-2}}\)
Using the model,
One limitation of the model is that it ignores air resistance.
| Scheme | Marks | AO |
|---|---|---|
| Use \(s = ut + \dfrac{1}{2}at^2\) vertically or any complete method to give an equation in \(t\) only | M1 | 3.4 |
| \(-70 = 65\sin\alpha \times t - \dfrac{1}{2} \times g \times t^2\) | A1 M(A)1 | 1.1b 1.1b |
| \(t = 7\) (s) | A1 | 1.1b |
| (4) |
Notes
Note that \(g = 10\); penalise once for whole question if \(g = 9.8\)
M1: Complete method, correct no. of terms, condone sign errors and sin/cos confusion
A1: Correct equation in \(t\) only with at most one error
M(A)1: Correct equation in \(t\) only
N.B. For ‘up and down’ methods etc, the two A marks are for all the equations that they use, lose a mark for each error.
A1: Cao (\(g = 9.8\), 7.1 or 7.11) (\(g = 9.81\), 7.1 or 7.12)
| Scheme | Marks | AO |
|---|---|---|
| Horizontal velocity component at \(A = 65\cos\alpha\) (60) | B1 | 3.4 |
| Complete method to find vertical velocity component at \(A\) | M1 | 3.4 |
| \(65\sin\alpha - g \times 7\) OR \(\sqrt{(-25)^2 + 2g \times 70}\) (45) | A1ft | 1.1b |
| Sub for trig and square, add and square root: \(\sqrt{60^2 + (-45)^2}\) | M1 | 3.1b |
| 75 Accept 80 \((\text{m s}^{-1})\) | A1 | 1.1b |
| (5) |
Notes
Note that \(g = 10\); penalise once for whole question if \(g = 9.8\)
B1: Seen, including on a diagram.
M1: Condone sign errors and sin/cos confusion
A1ft: Correct expression; accept negative of this, follow their \(t\)
M1: Sub for trig and use Pythagoras
A1: Cao (\(g = 9.8\) or 9.81, 75 or 74.8)
| Scheme | Marks | AO |
|---|---|---|
| e.g. an approximate value of \(g\) has been used, the dimensions of the stone could affect its motion, spin of the stone, \(g = 10\) instead of 9.8 has been used, \(g\) has been assumed to be constant, wind effect, shape of the stone | B1 | 3.5b |
| (1) | ||
| (10 marks) |
Notes
B1: B0 if incorrect extras