## Question:

The motor at C pulls in the cable with an acceleration $a_{C}$=$\left(3 t^{2}\right) \mathrm{m}/\mathrm{s}^{2}$ , where t is in seconds. The motor at D draws in its cable at $a_{D}$ = 5 m/${s}^{2}$ . If both motors start at the same instant from rest when d = 3 m , determine (a) the time needed for d = 0 , and (b) the relative velocity of block A with respect to block B when this occurs.

## Step-by-step

for A :

$s_{A}$+$\left(s_{A}-s_{C}\right)$=l\$
2 $v_{A}$=$v_{c}$
2 $a_{A}$=$a_{C}$=-3 $t^{2}$
$a_{A}$=-1.5 $t^{2}$=1.5 $t^{2}$ $\rightarrow$
$v_{A}$=0.5 $t^{3}$ $\rightarrow$
$s_{A}$=0.125 $t^{4}$ $\rightarrow$
For B:
$a_{B}$=5 m/${s}^{2}$
$v_{B}$=5 t
$s_{B}$=2.5 $t^{2}$
Require $s_{A}$+$s_{B}$=d
0.125 $t^{4}$+2.5 $t^{2}$=3
Set u=$t^{2}$     0.125 $u^{2}$+2.5 u=3
The positive root is u=1.1355 . Thus,
t=1.0656=1.07 s
$v_{A}$=0.5$(1.0656)^{3}$=0.6050
$v_{B}$=5(1.0656)=5.3281 m/s
${v}_{A}$=${v}_{B}$+${v}_{A / B}$
0.6050 i=-5.3281 i+$v_{A / B}$i
$v_{A / B}$=5.93 m/s $\rightarrow$