#### Answer

(a) At t = 0:
The particle's distance from the origin is zero.
At t = 2 s:
The particle's distance from the origin is 25.6 m
At t = 5 s:
The particle's distance from the origin is 160 m
(b) $v(t) = [(10.0\hat{i}+8.0\hat{j})~t] ~m/s$
(c) At t = 0:
The particle's speed is 0
At t = 2 s:
The particle's speed is 25.6 m/s
At t = 5 s:
The particle's speed is 64.0 m/s

#### Work Step by Step

$\vec{r} = (5.0\hat{i}+4.0\hat{j})~t^2 ~m$
(a) At t= 0:
$\vec{r} = (5.0\hat{i}+4.0\hat{j})~(0)^2 ~m$
$\vec{r} = 0$
The particle's distance from the origin is zero.
At t = 2 s:
$\vec{r} = (5.0\hat{i}+4.0\hat{j})~(2~s)^2 ~m$
$\vec{r} = (20.0\hat{i}+16.0\hat{j})~m$
We can find the distance from the origin.
$r = \sqrt{(20.0~m)^2+(16.0~m)^2}$
$r = 25.6~m$
The particle's distance from the origin is 25.6 m
At t = 5 s:
$\vec{r} = (5.0\hat{i}+4.0\hat{j})~(5~s)^2 ~m$
$\vec{r} = (125\hat{i}+100\hat{j})~m$
We can find the distance from the origin.
$r = \sqrt{(125~m)^2+(100~m)^2}$
$r = 160~m$
The particle's distance from the origin is 160 m
(b) $\vec{r} = (5.0\hat{i}+4.0\hat{j})~t^2 ~m$
$v(t) = \frac{dr}{dt}$
$v(t) = [(5.0\hat{i}+4.0\hat{j})~2t] ~m/s$
$v(t) = [(10.0\hat{i}+8.0\hat{j})~t] ~m/s$
(c) At t = 0:
$v = (10.0\hat{i}+8.0\hat{j})~(0) ~m/s$
$v = 0$
At t = 2 s:
$v = (10.0\hat{i}+8.0\hat{j})~(2~s)~m/s$
$v = (20.0\hat{i}+16.0\hat{j})~m/s$
We can find the speed.
$v = \sqrt{(20.0~m/s)^2+(16.0~m/s)^2}$
$v = 25.6~m/s$
The particle's speed is 25.6 m/s
At t = 5 s:
$v = (10.0\hat{i}+8.0\hat{j})~(5~s)~m/s$
$v = (50.0\hat{i}+40.0\hat{j})~m/s$
We can find the speed.
$v = \sqrt{(50.0~m/s)^2+(40.0~m/s)^2}$
$v = 64.0~m/s$
The particle's speed is 64.0 m/s