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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

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