9 A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = 2 t ^ { 4 } + k t ^ { 2 } - 4\).
The acceleration of \(P\) when \(t = 2\) is \(28 \mathrm {~ms} ^ { - 2 }\).
- Show that \(k = - 9\).
- Show that the velocity of \(P\) has its minimum value when \(t = 1.5\).
When \(t = 1 , P\) is at the point \(( - 6.4125,0 )\).
- Find the distance of \(P\) from the origin \(O\) when \(P\) is moving with minimum velocity.
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\(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is \(20 \mathrm {~m} . M\) is a point on the plane between \(A\) and \(B\). The surface of the plane is smooth between \(A\) and \(M\), and rough between \(M\) and \(B\).
A particle \(P\) is projected with speed \(4.2 \mathrm {~ms} ^ { - 1 }\) from \(A\) down the line of greatest slope (see diagram). \(P\) moves down the plane and reaches \(B\) with speed \(12.6 \mathrm {~ms} ^ { - 1 }\). The coefficient of friction between \(P\) and the rough part of the plane is \(\frac { \sqrt { 3 } } { 6 }\). - Find the distance \(A M\).
- Find the angle between the contact force and the downward direction of the line of greatest slope when \(P\) is in motion between \(M\) and \(B\).