Variable force (velocity v) - use v dv/dx

OCR M3 2007 June Q3

10 marks Standard +0.8

3 A particle $P$ of mass 0.2 kg is projected horizontally with speed $u \mathrm {~ms} ^ { - 1 }$ from a fixed point $O$ on a smooth horizontal surface. $P$ moves in a straight line and, at time $t \mathrm {~s}$ after projection, $P$ has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is $x \mathrm {~m}$ from $O$. The only force acting on $P$ has magnitude $0.4 v ^ { 2 } \mathrm {~N}$ and is directed towards $O$.

Show that $\frac { 1 } { v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 2$.
Hence show that $v = u \mathrm { e } ^ { - 2 x }$.
Find $u$, given that $x = 2$ when $t = 4$.

OCR M3 2014 June Q4

10 marks Standard +0.3

4 A particle $P$ of mass 0.4 kg is projected horizontally with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a fixed point $O$ on a smooth horizontal surface. At time $t \mathrm {~s}$ after projection $P$ is $x \mathrm {~m}$ from $O$ and is moving away from $O$ with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. There is a force of magnitude $1.6 v ^ { 2 } \mathrm {~N}$ resisting the motion of $P$.

Find an expression for $\frac { \mathrm { d } v } { \mathrm {~d} x }$ in terms of $v$, and hence show that $v = 2 \mathrm { e } ^ { - 4 x }$.
Find the distance travelled by $P$ in the 0.5 seconds after it leaves $O$.

Edexcel M4 2012 June Q3

16 marks Challenging +1.2

Two particles, of masses $m$ and $2 m$, are connected to the ends of a long light inextensible string. The string passes over a small smooth fixed pulley and hangs vertically on either side. The particles are released from rest with the string taut. Each particle is subject to air resistance of magnitude $k v ^ { 2 }$, where $v$ is the speed of each particle after it has moved a distance $x$ from rest and $k$ is a positive constant.
1. Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( v ^ { 2 } \right) + \frac { 4 k } { 3 m } v ^ { 2 } = \frac { 2 g } { 3 }$
2. Find $v ^ { 2 }$ in terms of $x$.
3. Deduce that the tension in the string, $T$, satisfies
$$\frac { 4 m g } { 3 } \leqslant T < \frac { 3 m g } { 2 }$$

OCR MEI M4 2007 June Q4

24 marks Challenging +1.8

4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ under the action of a force $F \mathrm {~N}$, where $F = 2 - 8 v ^ { 2 }$. The displacement of the particle from O at time $t$ seconds is $x \mathrm {~m}$.

Formulate and solve a differential equation to show that $v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)$.
Hence express $F$ in terms of $x$ and find, by integration, the work done in the first 2 m of the motion.
Formulate and solve a differential equation to show that $v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)$.
Calculate $v$ when $t = 1$ and when $t = 2$, giving your answers to four significant figures. Hence find the impulse of the force $F$ over the interval $1 \leqslant t \leqslant 2$.

Edexcel FM2 2019 June Q2

10 marks Challenging +1.2

A particle, $P$, of mass 0.4 kg is moving along the positive $x$-axis, in the positive $x$ direction under the action of a single force. At time $t$ seconds, $t > 0 , P$ is $x$ metres from the origin $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The force is acting in the direction of $x$ increasing and has magnitude $\frac { k } { v }$ newtons, where $k$ is a constant.

At $x = 3 , v = 2$ and at $x = 6 , v = 2.5$

Show that $v ^ { 3 } = \frac { 61 x + 9 } { 24 }$ The time taken for the speed of $P$ to increase from $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $T$ seconds.
Use algebraic integration to show that $T = \frac { 81 } { 61 }$

Edexcel FM2 2020 June Q3

10 marks Challenging +1.2

A particle $P$ of mass 0.5 kg is moving along the positive $x$-axis in the direction of $x$ increasing. At time $t$ seconds $( t \geqslant 0 ) , P$ is $x$ metres from the origin $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resultant force acting on $P$ is directed towards $O$ and has magnitude $k v ^ { 2 } \mathrm {~N}$, where $k$ is a positive constant.

When $x = 1 , v = 4$ and when $x = 2 , v = 2$

Show that $v = a b ^ { x }$, where $a$ and $b$ are constants to be found. The time taken for the speed of $P$ to decrease from $4 \mathrm {~ms} ^ { - 1 }$ to $2 \mathrm {~ms} ^ { - 1 }$ is $T$ seconds.
Show that $T = \frac { 1 } { 4 \ln 2 }$

OCR Further Mechanics 2024 June Q7

14 marks Standard +0.8

Show that $B$ 's motion can be modelled by the differential equation $\frac { 1 } { \mathrm { v } } \frac { \mathrm { dv } } { \mathrm { dx } } = - 4$.
1. Solve the differential equation in part (a) to find the particular solution for $v$ in terms of $x$ and $u$.
2. By considering the behaviour of $v$ as $x \longrightarrow \infty$ describe one feature of the model that is not realistic. At the instant when $B$ reaches the point $A$, where $\mathrm { x } = \mathrm { X }$, its speed is $V \mathrm {~ms} ^ { - 1 }$. The work done by the resistance as $B$ moves from $O$ to $A$ is denoted by $W \mathrm {~J}$.
1. Use the formula $\mathrm { W } = \int \mathrm { F } \mathrm { dx }$ to determine an expression for $W$ in terms of $X$ and $u$.
2. Explain the relevance of the sign of your answer in part (c)(i).
3. By writing your answer to part (c)(i) in terms of $V$ and $u$ show how the quantity $W$ relates to the energy of $B$.

OCR Further Mechanics 2023 June Q5

13 marks Challenging +1.3

A particle $P$ of mass $2$ kg moves along the $x$-axis. At time $t = 0$, $P$ passes through the origin $O$ with speed $3$ m s$^{-1}$. At time $t$ seconds the displacement of $P$ from $O$ is $x$ m and the velocity of $P$ is $v$ m s$^{-1}$, where $t \geqslant 0$, $x \geqslant 0$ and $v \geqslant 0$. While $P$ is in motion the only force acting on $P$ is a resistive force $F$ of magnitude $(v^2 + 1)$ N acting in the negative $x$-direction.

Find an expression for $v$ in terms of $x$. [5]
Determine the distance travelled by $P$ while its speed drops from $3$ m s$^{-1}$ to $2$ m s$^{-1}$. [2]

Particle $Q$ is identical to particle $P$. At a different time, $Q$ is moving along the $x$-axis under the influence of a single constant resistive force of magnitude $1$ N. When $t' = 0$, $Q$ is at the origin and its speed is $3$ m s$^{-1}$.

By comparing the motion of $P$ with the motion of $Q$ explain why $P$ must come to rest at some finite time when $t < 6$ with $x < 9$. [3]
Sketch the velocity-time graph for $P$. You do not need to indicate any values on your sketch. [1]
Determine the maximum displacement of $P$ from $O$ during $P$'s motion. [2]