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Calculus, EXT1* C1 2005 HSC 9a

A particle is initially at rest at the origin. Its acceleration as a function of time, `t`, is given by

`ddot x = 4sin2t`

  1. Show that the velocity of the particle is given by  `dot x = 2 − 2\ cos\ 2t`.  (2 marks)

  2. Sketch the graph of the velocity for  `0 ≤ t ≤ 2π`  AND determine the time at which the particle first comes to rest after  `t = 0`.  (3 marks)

  3. Find the distance travelled by the particle between  `t = 0`  and the time at which the particle first comes to rest after  `t = 0`.  (2 marks)

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  1. `text(See Worked Solutions)`
  2. `text(See Worked Solutions)`
  3. `2pi\ \ text(units)`
Show Worked Solution

i.   `text(Show)\ \ dot x = 2 − 2\ cos\ 2t`

`ddot x` `= 4\ sin\ 2t`
`dot x` `= int ddot x\ dt`
  `= int4\ sin\ 2t\ dt`
  `= −2\ cos\ 2t + c`

 
`text(When)\ t = 0, \ x = 0`

`0 = −2\ cos\ 0 + c`

`c = 2`

`:.dot x = 2 − 2\ cos\ 2t\ \ …text(as required)`

 

ii.  `text(Considering the range)`

`−1` `≤ \ \ \ cos\ 2t` `≤ 1`
`−2` `≤ \ \ \ 2\ cos\ 2t`  `≤ 2` 
`0` `≤ 2 − 2\ cos\ 2t` `≤ 4` 

 
`text(Period) = (2pi)/n = (2pi)/2 = pi`

Calculus in the Physical World, 2UA 2005 HSC 9a Answer

`text(After)\ t = 0,\ text(the particle next comes)`

`text(to rest at)\ t = pi.`

 

iii.  `text(Distance travelled)`

`= int_0^pi dot x\ dt`

`= int_0^pi 2 − 2\ cos\ 2t\ dt`

`= [2t − sin\ 2t]_0^pi`

`= [(2pi − sin\ 2pi) − (0 − sin\ 0)]`

`= 2pi\ \ text(units)`

Calculus, EXT1* C1 2010 HSC 7a

The acceleration of a particle is given by

`ddotx=4cos2t`,

where  `x`  is the displacement in metres and  `t`  is the time in seconds. 

Initially the particle is at the origin with a velocity of  `text(1 ms)^(–1)`.

  1. Show that the velocity of the particle is given by

     

      `dotx=2sin2t+1`.    (2 marks)

  2. Find the time when the particle first comes to rest.    (2 marks)

  3. Find the displacement,  `x`,  of the particle in terms of  `t`.    (2 marks)

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  1. `text{Proof  (See Worked Solutions)}`
  2. `(7pi)/12\ text(seconds)`
  3. `x=1-cos2t+t`
Show Worked Solution
i.   `text(Show)\ dotx` `=2sin2t+1`
`dotx` `=intddotx\ dt`
  `=int4cos2t\ dt`
  `=2sin2t+c`

 
`text(When)\ t=0, \ \ dotx=1\ \ text{(given)}`

`1=2sin0+c`

`c=1`

 
`:. dotx=2sin2t+1 \ \ \ text(… as required)`
 

ii.   `text(Find)\ t\ text(when)\ dotx=0 :`

`2sin2t+1` `=0`
`sin2t` `=-1/2`

 
`=>sin theta=1/2\ text(when)\ theta=pi/6`

`text(S)text(ince)\ \ sin theta\ \ text(is negative in 3rd and 4th quadrants)`

`2t` `=pi + pi/6`
`2t` `=(7pi)/6`
`t` `=(7pi)/12`

 
`:.\ text(Particle first comes to rest at)\ t=(7pi)/12\ text(seconds)`
 

iii.    `x` `=intdotx\ dt`
  `=int(2sin2t+1)\ dt`
  `=t-cos2t+c`

 
`text(When)\ t=0,\ x=0\ \ text{(given)}`

`0=0-cos0+c`

`c=1`

 
`:. x=t-cos2t+1`