A particle moves in a straight line such that its distance (x m) from the origin at t seconds is given by x = 45t + 11t2 − t3. Find the velocity of the particle after 9 seconds.

Here are the solutions to the problems.

9. Particle Motion and Curve Analysis

a) A particle moves in a straight line such that its distance ($x$ m) from the origin at $t$ seconds is given by $x = 45t + 11t^2 - t^3$.

• (i) Find the velocity of the particle after 9 seconds.

  Step 1: Find the velocity function by differentiating the distance function with respect to time $t$.

  $$v = \frac{dx}{dt} = \frac{d}{dt}(45t + 11t^2 - t^3) = 45 + 22t - 3t^2$$

  Step