The graph below represents the velocity map used to control an autonomous vehicle moving in a straight line for 40 seconds. Velocity vs time Velocity /m s o = 0 10 20 30 a0 50 Time /s (a) Find the following: i. the time at which the velocity is a maximum, ii. the displacement after 25 seconds, iii. the acceleration over the interval O to 10 seconds, iv. the interval over which the magnitude of the acceleration is a maximum and its value in m s (10 marks) (b) Draw a graph of the vehicle's acceleration with respect to time over the interval O to 40 seconds. (8 marks) (c) A sensor is attached to the rim of a vehicle's wheel to measure displacement in relation to a fixed point on the ground. The curve that describes the displacement of the sensor with respect to the fixed point is given by the functions x(t) = 6t + 3 cos(wt) and y(t) = 3 sin(wt) where w is a constant measured in rad s 1. Given that the velocity can be expressed as a vector in the form dx dy, V=gl + ) and the acceleration can be expressed as a vector in the form d?x d?y. a= 7l + ) find expressions for: i. the sensor's velocity vector, ii. the sensor's acceleration vector, iii. the magnitude of the acceleration in terms of w. (12 marks)