Velocity and acceleration problems

4 A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \mathrm {~s}\) after leaving \(O\), the displacement \(s \mathrm {~m}\) from \(O\) is given by \(s = t ^ { 3 } - 4 t ^ { 2 } + 4 t\) and the velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression for \(v\) in terms of \(t\).
2. Find the two values of \(t\) for which \(P\) is at instantaneous rest.
3. Find the minimum velocity of \(P\).

8. \begin{figure}[h] \end{figure} A car stops at two sets of traffic lights.
Figure 2 shows a graph of the speed of the car, \(v \mathrm {~ms} ^ { - 1 }\), as it travels between the two sets of traffic lights. The car takes \(T\) seconds to travel between the two sets of traffic lights.
The speed of the car is modelled by the equation $$v = ( 10 - 0.4 t ) \ln ( t + 1 ) \quad 0 \leqslant t \leqslant T$$ where \(t\) seconds is the time after the car leaves the first set of traffic lights.
According to the model,

1. find the value of \(T\)
2. show that the maximum speed of the car occurs when $$t = \frac { 26 } { 1 + \ln ( t + 1 ) } - 1$$ Using the iteration formula $$t _ { n + 1 } = \frac { 26 } { 1 + \ln \left( t _ { n } + 1 \right) } - 1$$ with \(t _ { 1 } = 7\)
3. 1. find the value of \(t _ { 3 }\) to 3 decimal places,
   2. find, by repeated iteration, the time taken for the car to reach maximum speed.

10
\includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.

1. Determine the values of \(p , q\) and \(r\).
2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).

5 A model car moves so that its distance, \(x\) centimetres, from a fixed point \(O\) after time \(t\) seconds is given by $$x = \frac { 1 } { 2 } t ^ { 4 } - 20 t ^ { 2 } + 66 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } x } { \mathrm {~d} t }\);
   2. \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } }\).
2. Verify that \(x\) has a stationary value when \(t = 3\), and determine whether this stationary value is a maximum value or a minimum value.
3. Find the rate of change of \(x\) with respect to \(t\) when \(t = 1\).
4. Determine whether the distance of the car from \(O\) is increasing or decreasing at the instant when \(t = 2\).

2 A bird flies from a tree. At time \(t\) seconds, the bird's height, \(y\) metres, above the horizontal ground is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - t ^ { 2 } + 5 , \quad 0 \leqslant t \leqslant 4$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\).
2. 1. Find the rate of change of height of the bird in metres per second when \(t = 1\).
   2. Determine, with a reason, whether the bird's height above the horizontal ground is increasing or decreasing when \(t = 1\).
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\) when \(t = 2\).
   2. Given that \(y\) has a stationary value when \(t = 2\), state whether this is a maximum value or a minimum value.