| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Mechanics (Further Paper 3 Mechanics) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Bungee jumping problems |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring careful geometric analysis of the rope configuration, energy conservation with elastic potential energy, and projectile motion after the ropes become slack. Part (a) involves multi-step energy calculations with non-trivial geometry (finding initial extension using Pythagoras), while part (b) requires determining velocity direction after a ballistic phase, demanding strong problem-solving skills beyond routine textbook exercises. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Initial Extension \(= \sqrt{50^2 + 20^2} - 30 = 23.85\) m | M1 | Uses Pythagoras to find the initial extension or the height to which Hannah rises |
| Initial EPE \(= 2 \times \frac{1}{2} \times \frac{3150}{30} \times 23.85^2 = 59735\) J | M1 | Calculates the initial EPE using their extension |
| Accept AWRT 59700 | A1 | Obtains correct initial EPE |
| GPE Gained \(= 84 \times 9.8 \times \left(50 - \sqrt{30^2 - 20^2}\right) = 84 \times 9.8 \times 27.639... = 22753\) J | A1 | Obtains correct GPE gained when ropes become slack. Accept AWRT 22750 |
| \(59735 - 22753 = \frac{1}{2} \times 84v^2\) | M1 | Uses conservation of energy to form an equation to find the speed |
| \(36982 = \frac{1}{2} \times 84v^2\) | ||
| \(v = \sqrt{880} = 29.6...\) | ||
| \(= 30 \text{ m s}^{-1}\) (2 sf) | R1 | Completes a rigorous argument and obtains correct speed giving answer to 2 sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Rope becomes taut at height \(h\): \(h = 50 + \sqrt{30^2 - 20^2} = 72.4\) metres | M1 | Obtains the first of two quantities that can be used as a basis for a comparison. e.g. (1) height at which rope becomes taut, (2) GPE when ropes become taut, (3) maximum height if ropes remain slack |
| Correct value obtained | A1 | Obtains the correct value for this quantity |
| \(59735 = 84 \times 9.8 \times h \Rightarrow h = 72.6\) metres | M1 | Obtains or states the second of two quantities for comparison. e.g. (1) maximum height if ropes remain slack, (2) initial EPE, (3) length of ropes at maximum height of 72.6 metres |
| Correct value obtained | A1 | Obtains the correct value for this quantity |
| The ropes just become taut at 72.6 metres above the bottom of the gorge, just below Hannah's highest point, at which point she is moving upwards | E1 | Infers, by comparing two correct values, that the ropes become taut just below Hannah's highest point, at which point she is moving upwards. Or infers that due to air resistance it is unlikely that the rope will become taut until she is moving down |
# Question 8(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Initial Extension $= \sqrt{50^2 + 20^2} - 30 = 23.85$ m | M1 | Uses Pythagoras to find the initial extension or the height to which Hannah rises |
| Initial EPE $= 2 \times \frac{1}{2} \times \frac{3150}{30} \times 23.85^2 = 59735$ J | M1 | Calculates the initial EPE using their extension |
| Accept AWRT 59700 | A1 | Obtains correct initial EPE |
| GPE Gained $= 84 \times 9.8 \times \left(50 - \sqrt{30^2 - 20^2}\right) = 84 \times 9.8 \times 27.639... = 22753$ J | A1 | Obtains correct GPE gained when ropes become slack. Accept AWRT 22750 |
| $59735 - 22753 = \frac{1}{2} \times 84v^2$ | M1 | Uses conservation of energy to form an equation to find the speed |
| $36982 = \frac{1}{2} \times 84v^2$ | | |
| $v = \sqrt{880} = 29.6...$ | | |
| $= 30 \text{ m s}^{-1}$ (2 sf) | R1 | Completes a rigorous argument and obtains correct speed giving answer to 2 sf |
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# Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Rope becomes taut at height $h$: $h = 50 + \sqrt{30^2 - 20^2} = 72.4$ metres | M1 | Obtains the first of two quantities that can be used as a basis for a comparison. e.g. (1) height at which rope becomes taut, (2) GPE when ropes become taut, (3) maximum height if ropes remain slack |
| Correct value obtained | A1 | Obtains the correct value for this quantity |
| $59735 = 84 \times 9.8 \times h \Rightarrow h = 72.6$ metres | M1 | Obtains or states the second of two quantities for comparison. e.g. (1) maximum height if ropes remain slack, (2) initial EPE, (3) length of ropes at maximum height of 72.6 metres |
| Correct value obtained | A1 | Obtains the correct value for this quantity |
| The ropes just become taut at 72.6 metres above the bottom of the gorge, just below Hannah's highest point, at which point she is moving upwards | E1 | Infers, by comparing two correct values, that the ropes become taut just below Hannah's highest point, at which point she is moving upwards. **Or** infers that due to air resistance it is unlikely that the rope will become taut until she is moving down |
**Total: 11 marks**8 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
A 'reverse' bungee jump consists of two identical elastic ropes. One end of each elastic rope is attached to either side of the top of a gorge.
The other ends are both attached to Hannah, who has mass 84 kg\\
Hannah is modelled as a particle, as shown in the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-12_467_844_678_598}
The depth of the gorge is 50 metres and the width of the gorge is 40 metres.\\
Each elastic rope has natural length 30 metres and modulus of elasticity 3150 N\\
Hannah is released from rest at the centre of the bottom of the gorge.\\
8
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of Hannah when the ropes become slack is $30 \mathrm {~ms} ^ { - 1 }$ correct to two significant figures.\\
8
\item Determine whether Hannah is moving up or down when the ropes become taut again. [5 marks]\\
\includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-14_2492_1721_217_150}
Additional page, if required.\\
Write the question numbers in the left-hand margin.
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\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Mechanics 2019 Q8 [11]}}