A girl operates a radio-controlled model car in a vacant parking lot. The girl's position is at the origin of the xy coordinate axes, and the surface of the parking lot lies in the x-y plane. The motion of the car is defined by the position vector r = (2 + 2t2)i + (6 + t3 )j where r and t are expressed in meters and seconds, respectively. Determine (a) the distance between the car and the girl when t = 2s, (b) the distance the car traveled in the interval from t = 0 to t = 2s, (c) the speed and direction of the car's velocity at t = 2s, (d) the magnitude of the car's acceleration at t = 2s.

a) 17.20

b) 11.31

c) 14.42

d) 12.65

Explanation:

(a)

The girl is at the origin of the x,y coordinates  (i.e 0,0,0  )

the position vector of the car at time 't' secs is

at t=2s,  the position vector is

Therefore, the the distance between the car and the girl is

s = 17.20

(b)

The position of the car at  t = 0s is

The position of the car at t = 2s is

The distance of the car traveled in the interval from t=0s to t=2 s is as follows:

(c)

The position vector of the car at time 't' secs is

The velocity of the car is

the direction of the car's velocity at t = 2s is  going to be

Thus; The speed of the car is

(d)    the car's acceleration is:

The magnitude of car's acceleration at t=2s is