Model and solve problems that involve definite integrals, including motion problems, with and without technology.

A particle moves in a straight line with velocity v(t) = 6 - 2t m/s, where t is time in seconds. Calculate the displacement of the particle from t = 0 to t = 3 seconds.

A particle moves along a straight line with velocity v(t) = 16t - 2t² (m s⁻¹), where t is time in seconds, for t ≥ 0. Calculate the displacement (m) of the particle from t = 0 to t = 4 seconds.

A particle moves along a straight line with velocity \( v(t) = 6t - t^2 \) m/s, where \( t \) is the time in seconds and \( 0 \leq t \leq 8 \). Explain how a definite integral can be used to determine the total distance travelled by the particle between \( t = 0 \) and \( t = 8 \) seconds, and state why the method differs from finding displacement.

A small drone is launched vertically from ground level. Its vertical velocity \( v(t) \) metres per second at time \( t \) seconds after launch is modelled by the function \[ v(t) = 12 - 0.6t^2 \] for \( 0 \le t \le 6 \). (a) Determine the height of the drone above ground level at \( t = 4 \) seconds. [2 marks] (b) Calculate the maximum height reached by the drone during the time interval \( 0 \le t \le 6 \) seconds, correct to one decimal place. [2 marks]

A person enters the lowest carriage of a swing ride. The ride moves vertically with simple harmonic motion. The vertical position, h metres, of the carriage above ground level at time t seconds is modelled by h(t) = 2.5 sin(πt/4) + 3.5 for t ≥ 0. It is claimed that: The maximum vertical acceleration experienced by riders on this swing ride exceeds 3 ms⁻². Evaluate the reasonableness of the claim.