Two smooth spheres \(A\) and \(B\), of masses \(2 m\) and \(m\) respectively, are connected by a light inextensible string which passes over a smooth fixed pulley as shown. \(A\) is initially at rest on the rough horizontal surface of a table, the coefficient of friction between \(A\)
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and the table being \(\frac { 2 } { 7 }\). \(B\) hangs freely on the end of the vertical portion of the string. \(A\) is now given an impulse, directed away from the pulley, of magnitude 5 m Ns.
Show that the system starts to move with speed \(2.5 \mathrm {~ms} ^ { - 1 }\).
State which modelling assumption ensures that the tensions in the two sections of the string can be taken to be equal.
Given that \(A\) comes to rest before it reaches the edge of the table and before \(B\) hits the pulley,
find the time taken for the system to come to rest.
Find the distance travelled by \(A\) before it first comes to rest.