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Mechanics, EXT2 M1 2024 HSC 16c

Two particles, \(A\) and \(B\), each have mass 1 kg and are in a medium that exerts a resistance to motion equal to \(k v\), where  \(k>0\)  and \(v\) is the velocity of any particle. Both particles maintain vertical trajectories.

The acceleration due to gravity is \(g\) ms\(^{-2}\), where  \(g>0\).

The two particles are simultaneously projected towards each other with the same speed, \(v_0\) ms\(^{-1}\), where  \(0<v_0<\dfrac{g}{k}\).

The particle \(A\) is initially \(d\) metres directly above particle \(B\), where  \(d<\dfrac{2 v_0}{k}\).

Find the time taken for the particles to meet.   (4 marks)

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\(t=-\dfrac{1}{k} \ln \left(\dfrac{2v_0-dk}{v_0}\right)\)

Show Worked Solution

\(\text{Particle A:}\)

\(\ddot{x}=\dfrac{dv_{\small{A}}}{dt}=-g-k v_{\small{A}}\)

\(\dfrac{dt}{dv_{\small{A}}}=\dfrac{1}{-g-kv_{\small{A}}}\)

\(t=-\displaystyle \int \dfrac{1}{g+k v_{\small{A}}} \, dv_{\small{A}}=-\dfrac{1}{k} \ln \left(g+k v_A\right)+c\)

\(\text{At} \ \ t=0, \ v_{\small{A}}=-v_0 \ \Rightarrow \ c=\dfrac{1}{k} \ln \left(g-k v_0\right)\)

\(t=\dfrac{1}{k} \ln \left(g-k v_0\right)-\dfrac{1}{k} \ln \left(g+kv_{\small{A}}\right)=\dfrac{1}{k} \ln \left(\dfrac{g-kv_0}{g+kv_{\small{A}}}\right)\)

\(\text{Find} \ v_{\small{A}}:\)

  \(\dfrac{g-k v_0}{g+k v_{\small{A}}}\) \(=e^{kt}\)
  \(g-kv_0\) \(=e^{kt} \cdot g+e^{kt} \cdot kv_{\small{A}}\)
  \(kv_{\small{A}}\) \(=e^{-kt}\left(g-kv_0\right)-g\)
  \(v_{\small{A}}\) \(=\dfrac{1}{k}\left[e^{-kt}\left(g-kv_0\right)-g\right]\)

  \(x\) \(=\displaystyle \frac{1}{k} \int e^{-kt} \cdot g-e^{-kt} \cdot v_0-g \, dt\)
    \(=\dfrac{1}{k}\left[-\dfrac{g}{k}e^{-kt}+v_0 e^{-kt}-gt\right]+c\)
    \(=-\left(\dfrac{g}{k^2}-\dfrac{v_0}{k}\right) e^{-kt}-gt+c\)

 
\(\text{When} \ \ t=0, x=0 \ \Rightarrow \ c=\dfrac{g}{k^2}-\dfrac{v_0}{k}\)

\(x_{\small{A}}=d-\left(\dfrac{g}{k^2}-\dfrac{v_0}{k}\right) e^{-kt}-gt+\dfrac{g}{k^2}-\dfrac{v_0}{k}\)
 

\(\text{Particle B:}\)

\(\ddot{x}=-g-k v_{\small{B}}\)

\(t=-\dfrac{1}{k} \ln \left(g+k v_{\small{B}}\right)+c\)
 

\(\text{When} \ \ t=0, v_B=v_0 \ \Rightarrow \ c=\dfrac{1}{k} \ln \left(g+kv_0\right)\)

  \(v_{\small{B}}\) \(=\dfrac{1}{k}\left[e^{-kt}\left(g+kv_0\right)-g\right]\)
  \(x_{\small{B}}\) \(=-\left(\dfrac{g}{k^2}+\dfrac{v_0}{k}\right) e^{-kt}-gt+c\)

 

\(\text{When} \ \ t=0, x=0 \ \Rightarrow \ c=\dfrac{g}{k^2}+\dfrac{v_0}{k}\)

\(x_{\small{B}}=-\left(\dfrac{g}{k^2}+\dfrac{v_0}{k}\right) e^{-kt}-gt+\dfrac{g}{k^2}+\dfrac{v_0}{k}\)
 

\(\text{Find  \(t\)  when \(\ x_{\small{A}}=x_{\small{B}}\):}\)

\(d-\left(\dfrac{g}{k^2}-\dfrac{v_0}{k}\right) e^{-k t}-g t+\dfrac{g}{k^2}-\dfrac{v_0}{k}=-\left(\dfrac{g}{k^2}+\dfrac{v_0}{k}\right) e^{-k t}-g t+\dfrac{g}{k^2}+\dfrac{v_0}{k}\)

  \(\dfrac{2 v_0}{k} \cdot e^{-k t}\) \(=\dfrac{2 v_0}{k}-d\)
  \(e^{-kt}\) \(=\left(\dfrac{2 v_0-d k}{k}\right) \cdot \dfrac{k}{v_0}\)
  \(-kt\) \(=\ln \left(\dfrac{2 v_0-d k}{v_0}\right)\)
  \(t\) \(=-\dfrac{1}{k} \ln \left(\dfrac{2v_0-dk}{v_0}\right)\)

Mechanics, EXT2 M1 2024 HSC 15c

A bar magnet is held vertically. An object that is repelled by the magnet is to be dropped from directly above the magnet and will maintain a vertical trajectory. Let \(x\) be the distance of the object above the magnet.
 

The object is subject to acceleration due to gravity, \(g\), and an acceleration due to the magnet \(\dfrac{27 g}{x^3}\), so that the total acceleration of the object is given by

 \(a=\dfrac{27 g}{x^3}-g\)

The object is released from rest at  \(x=6\).

  1. Show that  \(v^2=g\left(\dfrac{51}{4}-2 x-\dfrac{27}{x^2}\right)\).   (2 marks)

  2. Find where the object next comes to rest, giving your answer correct to 1 decimal place.  (2 marks)

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Show Worked Solution

i.    \(a=\dfrac{27 g}{x^3}-g\)

\(\dfrac{d}{dx}(\frac{1}{2}v^{2})\) \(= \dfrac{27g}{x^3}-g\)  
\(\dfrac{1}{2} v^2\) \(=-\dfrac{27 g}{2 x^2}-g x+c\)  

 
\(\text{When}\ \ x=6, v=0:\)

\(0\) \(=-\dfrac{27g}{2 \times 6^2}-6g+c\)  
\(c\) \(=\dfrac{459 g}{72}=\dfrac{51 g}{8}\)  

 

  \(\dfrac{1}{2} v^2\) \(=-\dfrac{27 g}{2 x^2}-g x+\dfrac{51 g}{8}\)
  \(v^2\) \(=-\dfrac{27 g}{x^2}-2 g x+\dfrac{51g }{4}\)
    \(=g\left(\dfrac{51}{4}-2 x-\dfrac{27}{x^2}\right)\)

 
ii.    \(\text{Find \(x\) when  \(v=0\):}\)

\(\dfrac{51}{4}-2 x-\dfrac{27}{x^2}\) \(=0\)  
\(51 x^2-8 x^3-108\) \(=0\)  
\(8 x^3-51 x^2+108\) \(=0\)  

 
\(\text{Given  \(x=6\)  is a root:}\)
 

\(8 x^3-51 x^2+108=(x-6)\left(8 x^2-3 x-18\right)\)

\(\text{Other roots:}\)

  \(x\) \(=\dfrac{3 \pm \sqrt{9-4 \cdot 8 \cdot 18}}{2 \times 8}\)
    \(=\dfrac{3 \pm \sqrt{585}}{16}\)
    \(=\dfrac{3+3 \sqrt{65}}{16} \quad(x>0)\)
    \(=1.7 \ \text{units (1 d.p.)}\)

 
\(\therefore \ \text{Object next comes to rest at  \(x=1.7\) units}\) 

Mechanics, EXT2 M1 2024 HSC 13c

A particle of unit mass moves horizontally in a straight line. It experiences a resistive force proportional to \(v^2\), where \(v\) m s\(^{-1}\) is the speed of the particle, so that the acceleration is given by  \(-k v^2\).

Initially the particle is at the origin and has a velocity of 40 m s\(^{-1}\) to the right. After the particle has moved 15 m to the right, its velocity is 10 m s\(^{-1}\) (to the right).

  1. Show that  \(v=40 e^{-k x}\).   (3 marks)

  2. Show that  \(k=\dfrac{\ln 4}{15}\).   (1 mark)

  3. At what time will the particle's velocity be 30 m s\(^{-1}\) to the right?   (3 marks)

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Show Worked Solution

i.    \(\ddot{x}=v \cdot \dfrac{d v}{d x}=-k v^2\)

  \(\dfrac{d v}{d x}\) \(=-k v\)
  \(\dfrac{d x}{d v}\) \(=-\dfrac{1}{k v}\)
  \(\displaystyle\int \frac{1}{v}\, d v\) \(=\displaystyle -\int k\, d x\)
  \(\ln \abs{v}\) \(=-k x+c\)
  \(v\) \(=e^{-k x+c}\)
    \(=A e^{-k x}\ \ (\text{where } A=e^c)\)

 
\(\text {When } x=0, v=40:\)

\(40=A e^{\circ} \ \Rightarrow \ A=40\)

\(\therefore V=40 \, e^{-k x}\)
 

ii.   \(\text {Show}\ \ k=\dfrac{\ln 4}{15}\)

\(\text {When } x=15, v=10:\)

  \(10\) \(=40 e^{-15 k}\)
  \(e^{-15 k}\) \(=\dfrac{1}{4}\)
  \(-15 k\) \(=\ln \left(\dfrac{1}{4}\right)\)
  \(15 k\) \(=\ln 4\)
  \(k\) \(=\dfrac{\ln 4}{15}\)

 

iii.   \(\text {Find}\ t\ \text {when}\ \ v=30:\)

  \(\dfrac{d v}{d t}\) \(=-k v^2\)
  \(\dfrac{d t}{d v}\) \(=-\dfrac{1}{k v^2}\)
  \(t\) \(=\displaystyle -\int \dfrac{1}{k v^2}\, d v=\dfrac{1}{k v}+c\)

 
\(\text {When}\ \ t=0, v=40:\)

\(0=\dfrac{1}{40 k}+c \ \Rightarrow \ c=-\dfrac{1}{40 k}\)
 

\(\text{Find \(t\) when  \(v=30\):}\)

  \(t\) \(=\dfrac{1}{30k}-\dfrac{1}{40k}\)
    \(=\dfrac{1}{120k}\)
    \(=\dfrac{15}{120\, \ln 4}\)
    \(=\dfrac{1}{8\,\ln 4}\  \text{seconds}\)

Mechanics, EXT2 M1 2020 HSC 14b

A particle starts from rest and falls through a resisting medium so that its acceleration, in m/s2, is modelled by

`a = 10 (1 - (kv)^2)`,

where `v` is the velocity of the particle in m/s and  `k = 0.01`.

Find the velocity of the particle after 5 seconds.  (4 marks)

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`46.21 \ text{ms}^-1`

Show Worked Solution
`a` `= 10 (1 – (kv)^2)`
`frac{dv}{dt}` `= 10 – 10 xx 0.01^2 xx v^2`
  `= 10 – 0.001 v^2`
`frac{dt}{dv}` `= frac{1}{10 – 0.001 \ v^2}`
  `= frac{1000}{10 \ 000 – v^2}`
`t` `= int frac{1000}{100^2 – v^2}\ dv`

 

`text{Using partial fractions}:`

`frac{1}{100^2 – v^2}` ` = frac{A}{100 + v} + frac{B}{100 – v}`
`1` `= A (100 – v) + B(100 + v)`

 
`text{If} \ \ v = 100 \ , \ 1 = 200 B \ => \ B = frac{1}{200}`

`text{If} \ \ v = -100 \ , \ 1 = 200 A \ => \ A = frac{1}{200}`
 

`t` `= 1000 int frac{1}{200 (100 + v)}\ dv + 1000 int frac{1}{200(100 -v)}\ dv`
  `= 5 int frac{1}{100 + v}\ dv +  5 int frac{1}{100 – v}\ dv`
  `= 5 ln \ | 100 + v | – 5 ln  \|100 – v | + c`
  `= 5 ln \ | frac{100 + v}{100 – v} | + c`

 
`text{When} \ \ t = 0 , \ v = 0`

`0 = 5 ln 1  + c  \ => \ c = 0`
 

`:. t = 5 ln \ | frac{100 + v}{100 – v} |`
  

`text{Find} \ \ v \ \ text{when} \ \ t = 5 :`

`5` `= 5 ln | frac{100 + v}{100 – v} |`
`1` `= ln | frac{100 + v}{100 – v} | `
`e` `= frac{100 + v}{100 – v}`
`100 e – ve` `= 100 + v`
`v + ve` `= 100 e – 100`
`v(1 + e)` `= 100 e – 100`
`:. v` `= frac{100 e – 100}{1 + e}`
  `= 46.21 \ text{ms}^-1 \ \ (text{2 d.p.})`

Mechanics, EXT2 M1 EQ-Bank 4

A torpedo with a mass of 80 kilograms has a propeller system that delivers a force of `F` on the torpedo, at maximum power. The water exerts a resistance on the torpedo proportional to the square of the torpedo's velocity `v`.

  1. Explain why  `(dv)/(dt) = 1/80 (F - kv^2)`
     
    where `k` is a positive constant.  (1 mark)

  2. If the torpedo increases its velocity from  `text(10 ms)\ ^(−1)`  to  `text(20 ms)\ ^(−1)`, show that the distance it travels in this time, `d`, is given by
     
         `d = 40/k log_e((F - 100k)/(F - 400k))`  (3 marks)

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  1. `text(See Worked Solutions)`
  2. `text(See Worked Solutions)`
Show Worked Solution

i.   `R ∝ v^2`

`R = −kv^2\ \ (k\ text{is positive constant})`

`text(Newton’s 2nd Law:)`

`text(Net Force)= mddotx` `= F – R`
`80ddotx` `= F – kv^2`
`ddotx` `= 1/80 (F – kv^2)`
`(dv)/(dt)` `= 1/80 (F – kv^2)`

 

ii.    `v · (dv)/(dx)` `= 1/80 (F – kv^2)`
  `(dv)/(dx)` `= (F – kv^2)/(80v)`
  `(dx)/(dv)` `= (80v)/(F – kv^2)`
  `x` `= −40 int (−2v)/(F – kv^2)\ dv`
    `= −40/k log_e (F – kv^2) + C`

 

`text(When)\ \ v = 10:`

`x_1 = −40/k log_e (F – 100k) + C`
 

`text(When)\ \ v = 20:`

`x_2 = −40/k log_e (F – 400k) + C`
 

`d` `= x_2 – x_1`
  `= −40/k log_e (F – 400k) + 40/k log_e (F – 100k)`
  `= 40/k log_e ((F – 100k)/(F – 400k))`

Mechanics, EXT2 M1 2015 HSC 15a

A particle  `A` of unit mass travels horizontally through a viscous medium. When  `t = 0`, the particle is at point  `O`  with initial speed  `u`. The resistance on particle  `A`  due to the medium is  `kv^2`, where  `v`  is the velocity of the particle at time  `t`  and  `k`  is a positive constant.

When  `t = 0`, a second particle  `B`  of equal mass is projected vertically upwards from  `O`  with the same initial speed  `u`  through the same medium. It experiences both a gravitational force and a resistance due to the medium. The resistance on particle  `B`  is  `kw^2`, where  `w`  is the velocity of the particle  `B`  at time  `t`. The acceleration due to gravity is  `g`.

  1. Show that the velocity  `v`  of particle  `A`  is given by  
     
         `1/v = kt + 1/u.`  (2 marks)

  2. By considering the velocity  `w`  of particle  `B`, show that
     
         `t = 1/sqrt(gk) (tan^-1(u sqrt(k/g)) - tan^-1 (w sqrt(k/g))).`  (3 marks)

  3. Show that the velocity  `V`  of particle  `A`  when particle  `B`  is at rest is given by
     
         `1/V = 1/u + sqrt(k/g) tan^-1 (u sqrt (k/g)).`  (1 mark)

  4. Hence, if  `u`  is very large, explain why  
     
         `V ~~ 2/pi sqrt(g/k).`  (1 mark)

Show Answers Only
  1. `text(Proof)\ \ text{(See Worked Solutions)}`
  2. `text(Proof)\ \ text{(See Worked Solutions)}`
  3. `text(Proof)\ \ text{(See Worked Solutions)}`
  4. `text(See Worked Solutions)`
Show Worked Solution

i.   `text(Particle)\ A:`

`ddot x` `= -kv^2`
`(dv)/(dt)` `= -kv^2`
`(dt)/(dv)` `=- 1/(kv^2)`
`t` `= -1/k int 1/v^2\ dv`
`-kt` `= -1/v + c`
`text(When)\ \ t=0,\ \ v=u\ \ \ \ =>c=1/u`
`-kt` `= -1/v + 1/u`
`:.1/v` `= kt + 1/u`

 

ii.   `text(Particle)\ B:`

`ddot x` `= -g – kw^2`
`(dw)/(dt)` `= -g – kw^2`
`(dt)/(dw)`   `=-1/(g + kw^2)`
`t`   `= – int (dw)/(g + kw^2)`
   `= -1/k int (dw)/(g/k + w^2)`
   `= -1/k xx 1/sqrt (g/k) tan^-1(w/sqrt (g/k)) + c`
  `= -1/sqrt (gk)\ tan^-1 ((sqrt k w)/sqrt g) + c`

 

`text(When)\ \ t = 0,\ \ w = u`

`=>c= 1/sqrt (gk) tan^-1 ((sqrt k u)/sqrt g)`

`:. t` `= -1/sqrt (gk) tan^-1 ((sqrt k w)/sqrt g) + 1/sqrt (gk) tan^-1 ((sqrt k u)/sqrt g)`
  `=1/sqrt (gk) (tan^-1 (u sqrt(k/g)) – 1/sqrt (gk) tan^-1 (w sqrt (k/g)))`

 

iii.   `B\ \ text(at rest when)\ \ w = 0`

`t = 1/sqrt (gk) (tan^-1 (u sqrt (k/g)))`

`:.1/V` `= k xx 1/sqrt(gk) tan^-1 (u sqrt (k/g)) + 1/u,\ \ \ \ \ text{(part (i))}`
  `= 1/u + sqrt(k/g) tan^-1 (u sqrt (k/g))`

 

iv.   `1/V = 1/u + sqrt (k/g) tan^-1 (u sqrt (k/g))`

`text(As)\ \ u -> oo,\ \ tan^-1 (u sqrt (k/g)) -> pi/2`

`:.\ text(If)\ \ u\ \ text(is very large,)`

`1/V` `~~ 0 + sqrt (k/g) xx pi/2`
`:.V` `~~ 2/pi sqrt (g/k)`

Mechanics, EXT2 M1 2012 HSC 13a

An object on the surface of a liquid is released at time  `t = 0`  and immediately sinks. Let  `x`  be its displacement in metres in a downward direction from the surface at time  `t`  seconds.

The equation of motion is given by

`(dv)/(dt) = 10 − (v^2)/40`,

where  `v`  is the velocity of the object.  

  1. Show that  `v = (20(e^t − 1))/(e^t + 1)`.   (4 marks)

  2. Use  `(dv)/(dt) = v (dv)/(dx)`  to show that  
     
         `x = 20\ log_e(400/(400 − v^2))`   (2 marks)

     

  3. How far does the object sink in the first 4 seconds?   (2 marks)

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  1. `text(See Worked Solutions.)`
  2. `text(See Worked Solutions.)`
  3. `40log_e((e^4 + 1)/(2e^2))\ text(m)`
Show Worked Solution
i.   `(dv)/(dt)` `= 10 − (v^2)/40`
  `(dv)/(dt)` `= (400 − v^2)/40`
  `dt` `= 40/(400 −v^2)\ dv`
  `int dt` `=int 40/(400 −v^2)\ dv`
  `t` `= int (1/(20 + v) + 1/(20 − v))\ dv`
    `= log_e(20 + v) − log_e(20 − v) + c`

 

`text(When)\ \ t = 0, v = 0\ \ => \ c = 0`

`t=` ` log_e((20 + v)/(20 − v))`
`e^t=` ` (20 + v)/(20 − v)`
`20e^t-ve^t=` ` 20 + v`
`v+ ve^t=` ` 20e^t − 20`
`v(1+e^t)=` `20(e^t − 1)`
`v=` ` (20(e^t − 1))/(e^t + 1)\ \ \ \ text(… as required)`

 

ii.   `v (dv)/(dx)` `= 10 − (v^2)/40`
  `(40v\ dv)/(400 − v^2)` `= dx`
`int dx` `= int (40v)/(400 − v^2)\ dv`
 `x` `= -20log_e(400 − v^2) + c`

 

`text(When)\ \ x = 0, v = 0\ \ \ => c = 20log_e 400`

`:.x` `= 20log_e400 − 20log_e(400 − v^2)`
  `= 20log_e((400)/(400 − v^2))\ \ \ \ text(… as required)`

 

iii.  `text(When)\ \ t = 4,\  v = (20(e^4 − 1))/(e^4 + 1)`

`x` `= 20log_e[400/(400 −((20(e^4 − 1))/(e^4 + 1))^2)]`
  `= 20log_e[((e^4 + 1)^2)/((e^4 + 1)^2 − (e^4 − 1)^2)]`
  `= 20log_e(((e^4 + 1)^2)/((e^4+1 + e^4 − 1)(e^4 + 1 − e^4 + 1)))`
  `= 20log_e(((e^4 + 1)^2)/(4e^4))`
  `= 40log_e\ (e^4 + 1)/(2e^2)\ \ text(metres)`