| Date | May 2011 | Marks available | 3 | Reference code | 11M.2.hl.TZ1.8 |
| Level | HL only | Paper | 2 | Time zone | TZ1 |
| Command term | Find | Question number | 8 | Adapted from | N/A |
Question
A jet plane travels horizontally along a straight path for one minute, starting at time \(t = 0\) , where \(t\) is measured in seconds. The acceleration, \(a\) , measured in ms−2, of the jet plane is given by the straight line graph below.
Find an expression for the acceleration of the jet plane during this time, in terms of \(t\) .
Given that when \(t = 0\) the jet plane is travelling at \(125\) ms−1, find its maximum velocity in ms−1 during the minute that follows.
Given that the jet plane breaks the sound barrier at \(295\) ms−1, find out for how long the jet plane is travelling greater than this speed.
Markscheme
equation of line in graph \(a = - \frac{{25}}{{60}}t + 15\) A1
\(\left( {a = - \frac{5}{{12}}t + 15} \right)\)
[1 mark]
\(\frac{{{\text{d}}v}}{{{\text{d}}t}} = - \frac{5}{{12}}t + 15\) (M1)
\(v = - \frac{5}{{24}}{t^2} + 15t + c\) (A1)
when \(t = 0\) , \(v = 125\) ms−1
\(v = - \frac{5}{{24}}{t^2} + 15t + 125\) A1
from graph or by finding time when \(a = 0\)
maximum \(= 395\) ms−1 A1
[4 marks]
EITHER
graph drawn and intersection with \(v = 295\) ms−1 (M1)(A1)
\(t = 57.91 - 14.09 = 43.8\) A1
OR
\(295 = - \frac{5}{{24}}{t^2} + 15t + 125 \Rightarrow t = 57.91...\); \(14.09...\)
\(t = 57.91... - 14.09... = 43.8\left( {8\sqrt {30} } \right)\) A1
[3 marks]
Examiners report
This question was well answered by a large number of candidates and indicated a good understanding of calculus, kinematics and use of the graphing calculator. Some candidates worked in \(x\) and \(y\) rather than \(a\), \(v\) and \(t\) but mostly obtained correct solutions. Although the majority of candidate used integration throughout the question some correct solutions were obtained by considering the areas in the diagram.
This question was well answered by a large number of candidates and indicated a good understanding of calculus, kinematics and use of the graphing calculator. Some candidates worked in \(x\) and \(y\) rather than \(a\), \(v\) and \(t\) but mostly obtained correct solutions. Although the majority of candidate used integration throughout the question some correct solutions were obtained by considering the areas in the diagram.
This question was well answered by a large number of candidates and indicated a good understanding of calculus, kinematics and use of the graphing calculator. Some candidates worked in \(x\) and \(y\) rather than \(a\), \(v\) and \(t\) but mostly obtained correct solutions. Although the majority of candidate used integration throughout the question some correct solutions were obtained by considering the areas in the diagram.
