Velocity from displacement function using calculus

6 The displacement of a particle is modelled by the equation \(\mathrm { s } = 7 + 4 \mathrm { t } - \mathrm { t } ^ { 2 }\), where \(s\) metres is the displacement from the origin at time \(t\) seconds. The diagram shows part of the displacement-time graph for the particle. The point \(( 2,11 )\) is the maximum point on the graph. \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_513_1381_422_255}

1. Kai argues that the point \(( 2,11 )\) is on the graph, so the particle has travelled a distance of 11 metres in the first 2 seconds. Comment on the validity of Kai's argument.
2. Determine the total distance the particle travels in the first 10 seconds.
3. Find an expression for the velocity of the particle at time \(t\).
4. Find the speed of the particle when \(t = 10\).

A particle moves in a straight line. At time \(t\) seconds, its displacement from a fixed point is \(s\) metres, where $$s = t^3 - 6t^2 + 9t$$

1. Find expressions for the velocity and acceleration of the particle at time \(t\). [4]
2. Find the times when the particle is at rest. [2]

The displacement, \(x\) m, from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36t + 3t^2 - 2t^3,$$ where \(t\) is the time in seconds and \(-4 \leqslant t \leqslant 6\).

1. Write down the displacement of the particle when \(t = 0\). [1]
2. Find an expression in terms of \(t\) for the velocity, \(v\) ms\(^{-1}\), of the particle. [2]
3. Find an expression in terms of \(t\) for the acceleration of the particle. [2]
4. Find the maximum value of \(v\) in the interval \(-4 \leqslant t \leqslant 6\). [3]
5. Show that \(v = 0\) only when \(t = -2\) and when \(t = 3\). Find the values of \(x\) at these times. [5]
6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [3]
7. Determine how many times the particle passes through O in the interval \(-4 \leqslant t \leqslant 6\). [3]