1.30: A particle moves along the -axis.

4PM1/2/November/2023 — Question 3 · 7 marks

A particle P\displaystyle P moves along the x\displaystyle x-axis.
At time t\displaystyle t seconds (t≥0\displaystyle t \geq 0) the acceleration, a\displaystyle am/s2\displaystyle \mathrm{m}/\mathrm{s}^{2}, of P\displaystyle P is given by a=6t−16\displaystyle a = 6t - 16
When t=0\displaystyle t = 0, P\displaystyle P is at the origin and is moving with velocity 12 m/s\displaystyle \mathrm{m}/\mathrm{s}.
(a) Find an expression in terms of t\displaystyle t for
(i) the velocity of P\displaystyle P at time t\displaystyle t seconds
(ii) the displacement of P\displaystyle P at time t\displaystyle t seconds.
(4)
(b) Hence find the time at which P\displaystyle P first returns to the origin.
(3)