Determine displacement given acceleration and initial values of displacement and vel ocity.

A particle moves along a straight line with acceleration \( a(t) = 6t - 4 \) m/s², where \( t \) is the time in seconds. At \( t = 0 \), the particle has velocity \( v(0) = 3 \) m/s and displacement \( x(0) = 2 \) m. The displacement of the particle at \( t = 2 \) seconds is

A model rocket is launched vertically upward from a platform. The rocket's acceleration (m s⁻²) is given by \( a(t) = 12 - 3t \) for \( t \geq 0 \), where \( t \) is the time (s) since launch. At launch (\( t = 0 \)), the rocket is 5.0 m above the ground with an initial velocity of 8.0 m s⁻¹ upward. Determine the rocket's displacement from the ground at \( t = 4.0 \) s.

A drone is performing a vertical test flight along a straight-line path. The drone's acceleration (m s⁻²) during the flight is modelled by \( a(t) = 6\cos\left(\frac{\pi t}{4}\right) - 2 \), where \( t \) is time (s) for \( 0 \le t \le 8 \). The drone begins its flight from ground level (displacement = 0 m) with an initial velocity of 5 m s⁻¹ upward. Determine the displacement of the drone from ground level when \( t = 6 \) seconds.