Question

The position vector r describes the path of an object moving in the xy-plane. position vector point r(t) = ti + (ât2 + 7)j (1, 6) (a) find the velocity vector, speed, and acceleration vector of the object.

Answer 1

We can decompose the problem on x- and y-axis.

The position vector decomposed is:
$r_x = t$
$r_y = at^2 + 7$

The velocity vector can be found computing the derivative of r on both axes:
$r'_x = 1$
$r'_y=2at$
So, the velocity vector is
r' = 1i+2atj

The speed (the magnitude of the velocity vector) is
$v= \sqrt{(1)^2+(2at)^2}$

Finally we can write the acceleraion vector by performing derivation on the velocity vector:
$r''_x=0$
$r''_y=2a$
and so
r''=2a j