A car is travelling along a straight, flat road. The velocity, \(v\) km h\(^{-1}\), of the car and its position, \(x\) kilometres, are measured from the position on the road where  \(x=0\).

The velocity \(v\) and the position \(x\) of the car are related by  \(v^2=1600+\dfrac{672}{\pi} \arccos \left(\dfrac{x}{20}\right)\), where  \(-15 \leq x \leq 15\)  and  \(v \geq 0\).

A speed detection device is positioned to detect the speed of a car as it passes the position  \(x=0\). The speed limit on the road is 40 km h\(^{-1}\).

The speed detection device will be activated if the car is travelling at 10% or more above the speed limit.

1. Determine, with evidence, whether the speed detection device will be activated.   (1 mark)
   

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2. Find the acceleration of the car, in km h\(^{-2}\), when  \(x=12\).
3. Give your answer in the form  \(\dfrac{k}{\pi}\), where  \(k \in Z\).   (3 marks)
   

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A body moves in a straight line so that when its displacement from a fixed origin `O` is `x` metres, its acceleration, `a`, is `-4 x \ text{ms}^{-2}`. The body accelerates from rest and its velocity, `v`, is equal to `-2 \ text{ms}^{-1}` as it passes through the origin. The body then comes to rest again.

Find `v` in terms of `x` for this interval.  (4 marks)

A particle moves along a straight line. When the particle is \(x\) m from a fixed point \(O\), its velocity, \( v\) m s\(^{-1}\), is given by

\(v=\dfrac{3 x+2}{2 x-1}\), where  \(x \geq 1\).

1. Find the acceleration of the particle, in m s\(^{-2}\), when  \(x=2\).   (2 marks)

2. Find the value that the velocity of the particle approaches as \(x\) becomes very large.   (1 mark)

The velocity, `v` ms`\ ^(−1)`, of a particle at time  `t >= 0`  seconds and at position  `x >= 1`  metre from the origin is  `v = 1/x`.

The acceleration of the particle, in `text(ms)^(−2)`, when  `x = 2`  is

1. `−1/4`
2. `−1/8`
3. `1/8`
4. `1/2`
5. `1/4`

A variable force acts on a particle, causing it to move in a straight line. At time `t` seconds, where  `t >= 0`, its velocity `v` metres per second and position `x` metres from the origin are such that  `v = e^x sin(x)`.

The acceleration of the particle, in ms⁻², can be expressed as

1. `e^(2x)(sin^2(x) + 1/2sin(2x))`
2. `e^x sin(x)(sin(x) + cos(x))`
3. `e^x(sin(x) + cos(x))`
4. `1/2 e^(2x) sin^2(x)`
5. `e^x cos(x)`

A body is moving in a straight line and, after  `t`  seconds, it is `x` metres from the origin and travelling at  `v` ms`\ ^(−1)`.

Given that  `v = x`, and that  `t = 3`  where  `x = −1`, the equation for `x` in terms of  `t`  is

1. `x = e^(t - 3)`
2. `x = −e^(3 - t)`
3. `x = sqrt(2t - 5)`
4. `x = −sqrt(2t - 5)`
5. `x = −e^(t - 3)`

A particle moves in a straight line. At time  `t`  seconds, where  `t >= 0`, its displacement `x` metres from the origin and its velocity  `v`  metres per second are such that  `v = 25 + x^2`.

If  `x = 5`  initially, then  `t`  is equal to

1. `25x + (x^3)/3`
2. `25x + (x^3)/3 - 500/3`
3. `1/5tan^(−1)(x/5) + 5`
4. `tan^(−1)(x/5) - pi/4`
5. `1/5tan^(−1)(x/5) - pi/20`

A body moves in a straight line such that its velocity  `v\ text(ms)^(-1)`  is given by  `v = 2sqrt(1 - x^2)`, where `x` metres is its displacement from the origin.

The acceleration of the body in `text(ms)^(-2)`  is given by

1. `(−2x)/(sqrt(1 - x^2))`
2. `−2x`
3. `2/(sqrt(1 - x^2))`
4. `2(1 - 2x)`
5. `−4x`

The velocity, `v` m/s, of a body when it is `x` metres from a fixed point `O` is given by

`v = (2x)/sqrt(1 + x^2).`

Find an expression for the acceleration of the body in terms of `x` in simplest form.  (3 marks)

The acceleration, in `text(ms)^(-2)`, of a particle moving in a straight line is given by  `–4x`, where `x` metres is its displacement from a fixed origin `O`.

If the particle is at rest where  `x = 5`, the speed of the particle, in `text(ms)^(−1)`, where  `x = 3`  is

A.           `8`

B.     `8 sqrt 2`

C.         `12`

D.     `4 sqrt 2`

E.   `2 sqrt 34`