Kinematics: displacement-velocity-acceleration

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 \(t = 2\).

A cyclist is riding a bicycle along a straight horizontal road \(AB\) of length 50 m. The cyclist starts from rest at \(A\) and reaches a speed of \(6 \text{ m s}^{-1}\) at \(B\). The cyclist produces a constant driving force of magnitude 100 N. There is a resistance force, and the work done against the resistance force from \(A\) to \(B\) is 3560 J. Find the total mass of the cyclist and bicycle. [3]

A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the top of the plane is 2 m s\(^{-1}\) and the speed of the block at the bottom of the plane is 4 m s\(^{-1}\). Find the work done against the resistance to motion of the block. [4]

A child is playing with a small model of a fire-engine of mass 0.5 kg and a straight, rigid plank. The plank is inclined at an angle \(\alpha\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(\alpha = 20°\) the fire-engine is projected with an initial speed of 5 m s\(^{-1}\) and first comes to rest after travelling 2 m.

1. Find, to 3 significant figures, the value of \(R\). [7]

When \(\alpha = 40°\) the fire-engine is again projected with an initial speed of 5 m s\(^{-1}\).

1. Find how far the fire-engine travels before first coming to rest. [3]

A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \text{m s}^{-1}\), at time \(t\) seconds, is modelled by the equation $$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$

1. Find the distance the car travels during the first 10 seconds of its journey. [3 marks]
2. Find the maximum speed of the car. Give your answer to three significant figures. [4 marks]
3. Deduce the range of values of \(t\) for which the car is modelled as decelerating. [2 marks]

A student is attempting to model the flight of a boomerang. She throws the boomerang from a fixed point \(O\) and catches it when it returns to \(O\). She suggests the model for the displacement, \(s\) metres, after \(t\) seconds is given by \(s = 9t^2 - \frac{3}{2}t^3\), \(0 \leq t \leq 6\). For this model,

1. determine what happens at \(t = 6\), [2]
2. find the greatest displacement of the boomerang from \(O\), [4]
3. find the velocity of the boomerang 1 second before the student catches it, [2]
4. find the acceleration of the boomerang 1 second before the student catches it. [2]