OCR MEI M4 2009 June — Question 4 24 marks

Exam BoardOCR MEI
ModuleM4 (Mechanics 4)
Year2009
SessionJune
Marks24
PaperDownload PDF ↗
TopicVariable Force
TypeAir resistance kv² - falling from rest or projected downward
DifficultyChallenging +1.2 This is a standard M4/FM mechanics question on variable force with air resistance. Parts (i) and (iv) are routine applications of terminal velocity and given formulas. Part (ii) requires separating variables and integrating, which is standard technique. Part (iii) involves work-energy calculations that are bookwork for this module. Part (v) requires formulating and solving a differential equation, which is expected at this level. While multi-part and requiring several techniques, all steps follow standard M4 procedures without requiring novel insight.
Spec6.02b Calculate work: constant force, resolved component6.02c Work by variable force: using integration6.06a Variable force: dv/dt or v*dv/dx methods
4 A parachutist of mass 90 kg falls vertically from rest. The forces acting on her are her weight and resistance to motion \(R \mathrm {~N}\). At time \(t \mathrm {~s}\) the velocity of the parachutist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance she has fallen is \(x \mathrm {~m}\). While the parachutist is in free-fall (i.e. before the parachute is opened), the resistance is modelled as \(R = k v ^ { 2 }\), where \(k\) is a constant. The terminal velocity of the parachutist in free-fall is \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = \frac { g } { 40 }\).
  2. Show that \(v ^ { 2 } = 3600 \left( 1 - \mathrm { e } ^ { - \frac { g x } { 1800 } } \right)\). When she has fallen 1800 m , she opens her parachute.
  3. Calculate, by integration, the work done against the resistance before she opens her parachute. Verify that this is equal to the loss in mechanical energy of the parachutist. As the parachute opens, the resistance instantly changes and is now modelled as \(R = 90 v\).
  4. Calculate her velocity just before opening the parachute, correct to four decimal places.
  5. Formulate and solve a differential equation to calculate the time it takes after opening the parachute to reduce her velocity to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4 A parachutist of mass 90 kg falls vertically from rest. The forces acting on her are her weight and resistance to motion $R \mathrm {~N}$. At time $t \mathrm {~s}$ the velocity of the parachutist is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the distance she has fallen is $x \mathrm {~m}$.

While the parachutist is in free-fall (i.e. before the parachute is opened), the resistance is modelled as $R = k v ^ { 2 }$, where $k$ is a constant. The terminal velocity of the parachutist in free-fall is $60 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(i) Show that $k = \frac { g } { 40 }$.\\
(ii) Show that $v ^ { 2 } = 3600 \left( 1 - \mathrm { e } ^ { - \frac { g x } { 1800 } } \right)$.

When she has fallen 1800 m , she opens her parachute.\\
(iii) Calculate, by integration, the work done against the resistance before she opens her parachute. Verify that this is equal to the loss in mechanical energy of the parachutist.

As the parachute opens, the resistance instantly changes and is now modelled as $R = 90 v$.\\
(iv) Calculate her velocity just before opening the parachute, correct to four decimal places.\\
(v) Formulate and solve a differential equation to calculate the time it takes after opening the parachute to reduce her velocity to $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

\hfill \mbox{\textit{OCR MEI M4 2009 Q4 [24]}}
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