M1 June 2015 Q7
7. A train travels along a straight horizontal track between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration 0.5 m s\(^{-2}\) until it reaches a speed of \(V\) m s\(^{-1}\), \((V < 50)\). The train then travels at this constant speed before it moves with constant deceleration 0.25 m s\(^{-2}\) until it comes to rest at \(B\).
The total time for the journey from \(A\) to \(B\) is 5 minutes.
Given that the distance between the two stations \(A\) and \(B\) is 6.3 km,
| Scheme | Marks |
|---|---|
| B1 (shape) | |
| B1 (\(V\)) | |
| (2) |
Notes
B1 for a trapezium with line starting and finishing on the \(t\)-axis
B1 for \(V\) correctly marked
| Scheme | Marks |
|---|---|
| (i) (ii) \(\dfrac{V}{t_1} = \dfrac{1}{2} \Rightarrow t_1 = 2V\) s; \(\ t_2 = 4V\) s | M1 A1; A1 |
| (iii) \(t_3 = 300 - 2V - 4V = 300 - 6V\) s | M1 A1 |
| (5) |
Notes
First M1 for a correct method
First A1 for \(V/0.5\) oe
Second A1 for \(V/0.25\) oe
Second M1 for (300 – sum of previous answers) Allow 5 instead of 300.
Third A1 for \(300 - 6V\) oe
| Scheme | Marks |
|---|---|
| \(6300 = \dfrac{V(300 + 300 - 6V)}{2}\) or \(\dfrac{1}{2}2V.V + (300 - 6V).V + \dfrac{1}{2}4V.V\) | M1 A1 ft |
| \(V^2 - 100V + 2100 = 0\) | A1 |
| \((V - 30)(V - 70) = 0\) | M1 A1 |
| \(V = 30\) or 70 | |
| \(V = 30\ (< 50)\) | A1 |
| (6) | |
| (13 marks) |
Notes
First M1 for using the area under the curve (distance travelled) to form an equation in \(V\) only. (Allow use of 6.3 but must see ½ used at least once in their expression.)
First A1 ft on their answers in (b) for a correct equation so must have used 6300 not 6.3
Second A1 for correct equation in form \(aV^2 + bV + c = 0\)
Second M1 for solving a 3 term quadratic. (Can be implied by correct answers)
Second A1 for either 30 or 70
Third A1 for 30 as final answer.
N.B. If answer(s) are wrong or have come from an incorrect quadratic, and the quadratic formula is used, M1 can only be awarded if there is clear evidence that the correct formula has been used. i.e. we need to see numbers substituted into a stated correct formula.