The acceleration of a particle \(P\) is \(( 8 t - 18 ) \mathrm { ms } ^ { - 2 }\), where \(t\) seconds is the time that has elapsed since \(P\) passed through a fixed point \(O\) on the straight line on which it is moving.
At time \(t = 3 , P\) has speed \(2 \mathrm {~ms} ^ { - 1 }\). Find
- the velocity of \(P\) at time \(t\),
- the values of \(t\) when \(P\) is instantaneously at rest.
- A pump raises water from a reservoir at a depth of 25 m below ground level. The water is delivered at ground level with speed \(12 \mathrm {~ms} ^ { - 1 }\) through a pipe of radius 4 cm . Find
- the potential and kinetic energy given to the water each second,
- the rate, in kW , at which the pump is working.
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[ \(1 \mathrm {~m} ^ { 3 }\) of water has a mass of 1000 kg .] - A particle \(P\) of mass 3 kg has position vector \(\mathbf { r } = \left( 2 t ^ { 2 } - 4 t \right) \mathbf { i } + \left( 1 - t ^ { 2 } \right) \mathbf { j } \mathbf { m }\) at time \(t\) seconds.
- Find the velocity vector of \(P\) when \(t = 3\).
- Find the magnitude of the force acting on \(P\), showing that this force is constant.
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The diagram shows a uniform lamina \(A B C D E\) formed by removing a symmetrical triangular section from a rectangular sheet of metal measuring 30 cm by 25 cm .