on 10

The graph below represents the velocity map used to control an autonomous

vehicle moving in a straight line for 40 seconds.

Velocity /m s-¹

140

120

100

80

60

40

20

0

O

Velocity vs time

10

(a) Find the following:

i.

ii.

iii.

iv.

20

30

Time /s

40

50

the time at which the velocity is a maximum,

the displacement after 25 seconds,

the acceleration over the interval 0 to 10 seconds,

the interval over which the magnitude of the acceleration is a maximum

and its value in m s-².

(10 marks)

(b) Draw a graph of the vehicle's acceleration with respect to time over the

interval 0 to 40 seconds.

(8 marks)/n(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

dt dt

and the acceleration can be expressed as a vector in the form

d²x d²y

i+

dt²

dt²

ii.

v==i+

find expressions for:

i.

iii.

a =

the sensor's velocity vector,

the sensor's acceleration vector,

the magnitude of the acceleration in terms of w.

(12 marks)