M1 June 2006 Q3
3. A train moves along a straight track with constant acceleration. Three telegraph poles are set at equal intervals beside the track at points \(A\), \(B\) and \(C\), where \(AB = 50\) m and \(BC = 50\) m. The front of the train passes \(A\) with speed 22.5 m s\(^{-1}\), and 2 s later it passes \(B\). Find
| Scheme | Marks |
|---|---|
| \(AB\): \(50 = 2 \times 22.5 + \tfrac{1}{2}a.4\) | M1 A1 |
| \(\Rightarrow a = 2.5\) m s\(^{-2}\) | A1 |
| (3) |
Notes
NB note slight changes to scheme: dependency now in (c) and new rule on accuracy of answers.
| Scheme | Marks |
|---|---|
| \(v^2 = 22.5^2 + 2 \times 2.5 \times 100\) | M1 A1ft |
| \(\Rightarrow v \approx 31.7(2)\) m s\(^{-1}\) | A1 |
| (3) |
Notes
(b) M1 for valid use of data (e.g. finding speed at \(B\) by spurious means and using this to get \(v\) at \(C\) is M0.
Accept answer as AWRT 31.7
In (b) and (c), f.t. A marks are for f.t. on wrong \(a\) and/or answer from (b).
| Scheme | Marks |
|---|---|
| \(v_B = 22.5 + 2 \times 2.5 = 27.5\) (must be used) | M1 |
| \(31.72 = 27.5 + 2.5t\) OR \(50 = 27.5t + \tfrac{1}{2} \times 2.5t^2\) OR \(50 = \tfrac{1}{2}(27.5 + 31.72)t\) | M1 A1ft |
| \(\Rightarrow t \approx 1.69\) s | A1 |
| (4) | |
| (10 marks) |
Notes
OR
| Scheme | Marks |
|---|---|
| \(31.72 = 22.5 + 2.5T\) OR \(100 = 22.5T + \tfrac{1}{2} \times 2.5T^2\) | M1 A1ft |
| \(\Rightarrow T \approx 3.69\) | |
| \(\Rightarrow t \approx 3.69 - 2 = 1.69\) s | M1 A1 |
| (4) |
OR
| Scheme | Marks |
|---|---|
| \(50 = 31.7t - \tfrac{1}{2} \times 2.5t^2\) | M2 A1ft |
| Solve quadratic to get \(t = 1.69\) s | A1 |
| (4) |
(c) M1 + M1 to get to an equation in the required \(t\) (normally two stages, but they can do it in one via 3rd alternative above)
Ans is cao. Hence premature approx (→ e.g. 1.68) is A0.
But if they use a 3 sf answer from (b) and then give answer to (c) as 1.7, allow full marks. And accept 2 or 3 s.f. answer or better to (c).
(Corrected from the printed mark scheme: in the first OR the second equation is printed as \(100 = 22.5t + \tfrac{1}{2} \times 2.5T^2\).)