OCR MEI M4 2009 June — Question 1 12 marks

Exam BoardOCR MEI
ModuleM4 (Mechanics 4)
Year2009
SessionJune
Marks12
PaperDownload PDF ↗
TopicVariable Force
TypeVariable mass problems (mass increasing)
DifficultyChallenging +1.8 This is a challenging M4 variable mass problem requiring application of the rocket equation d(mv)/dt = F, algebraic manipulation to reach a non-standard differential equation form, integration involving partial fractions or substitution, and physical interpretation. It demands multiple sophisticated techniques beyond standard A-level, though the steps are guided.
Spec6.03b Conservation of momentum: 1D two particles6.06a Variable force: dv/dt or v*dv/dx methods
1 A raindrop increases in mass as it falls vertically from rest through a stationary cloud. At time \(t \mathrm {~s}\) the velocity of the raindrop is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its mass is \(m \mathrm {~kg}\). The rate at which the mass increases is modelled as \(\frac { m g } { 2 ( v + 1 ) } \mathrm { kg } \mathrm { s } ^ { - 1 }\). Resistances to motion are neglected.
  1. Write down the equation of motion of the raindrop. Hence show that $$\left( 1 - \frac { 1 } { v + 2 } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 2 } g .$$
  2. Solve this differential equation to find an expression for \(t\) in terms of \(v\). Calculate the time it takes for the velocity of the raindrop to reach \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Describe, with reasons, what happens to the acceleration of the raindrop for large values of \(t\).
1 A raindrop increases in mass as it falls vertically from rest through a stationary cloud. At time $t \mathrm {~s}$ the velocity of the raindrop is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its mass is $m \mathrm {~kg}$. The rate at which the mass increases is modelled as $\frac { m g } { 2 ( v + 1 ) } \mathrm { kg } \mathrm { s } ^ { - 1 }$. Resistances to motion are neglected.\\
(i) Write down the equation of motion of the raindrop. Hence show that

$$\left( 1 - \frac { 1 } { v + 2 } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 2 } g .$$

(ii) Solve this differential equation to find an expression for $t$ in terms of $v$. Calculate the time it takes for the velocity of the raindrop to reach $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(iii) Describe, with reasons, what happens to the acceleration of the raindrop for large values of $t$.

\hfill \mbox{\textit{OCR MEI M4 2009 Q1 [12]}}
This paper (4 questions)
View full paper
Q1 12 Q2 12 Q3 24 Q4 24