7 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(\frac { 200 } { x ^ { 2 } } - \frac { 100 } { x ^ { 3 } }\) for \(x > 0\). When \(t = 0 , x = 1\) and \(P\) has velocity \(10 \mathrm {~ms} ^ { - 1 }\) directed towards \(O\).

1. Show that the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) is given by \(\mathrm { v } = \frac { 10 ( 1 - 2 \mathrm { x } ) } { \mathrm { x } }\).
2. Show that \(x\) and \(t\) are related by the equation \(\mathrm { e } ^ { - 40 \mathrm { t } } = ( 2 \mathrm { x } - 1 ) \mathrm { e } ^ { 2 \mathrm { x } - 2 }\) and deduce what happens to \(x\) as \(t\) becomes large.
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