Construct and interpret displacement-time graphs, with velocity as the slope of the tangent.

A particle moves along a straight line. Its displacement from a fixed origin is given by \( s(t) = 2\ln(3 + \cos(t)) + 4 \), where \( s \) is measured in metres and \( t \) is time in seconds, \( t \ge 0 \). Use calculus methods to determine the velocity of the particle at \( t = \frac{\pi}{3} \) seconds.

A robot moves along a straight track during a timed trial. Its displacement \(s\) (metres) from the starting point at time \(t\) (seconds) is recorded at regular intervals, as shown in the table below. (a) Using the data, determine the average velocity of the robot during the time interval from \(t = 2\) to \(t = 8\) seconds. (1 mark) (b) By constructing tangent approximations at \(t = 4\) and \(t = 10\), estimate the instantaneous velocity of the robot at each of these times. (2 marks) (c) Determine the time interval during which the robot's velocity is greatest. Justify your answer by reference to the slope of the displacement-time relationship. (2 marks)

A particle moves along a straight line. Its displacement in metres from a fixed point is given by $s(t) = 4t^2 - 12t + 5$, where $t$ is the time in seconds. (a) Find the displacement of the particle at $t = 0.5$ seconds. [1] (b) The velocity of the particle is the rate of change of displacement with respect to time. Determine the velocity at $t = 2$ seconds. [1] (c) Explain what the sign of the velocity at $t = 2$ seconds tells us about the direction of motion of the particle. [1]