6 A particle \(P\) of mass 0.1 kg moves with decreasing speed in a straight line on a smooth horizontal surface. A horizontal resisting force of magnitude \(0.2 \mathrm { e } ^ { - x } \mathrm {~N}\) acts on \(P\), where \(x \mathrm {~m}\) is the displacement of \(P\) from a fixed point \(O\) on the line. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement from \(O\) is \(x \mathrm {~m}\).

1. Show that $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = k \mathrm { e } ^ { - x }$$ where \(k\) is a constant to be found.
   \(P\) passes through \(O\) with velocity \(2.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate the value of \(x\) at the instant when the velocity of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Show that the speed of \(P\) does not fall below \(0.917 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
   [0pt] [Question 7 is printed on the next page.]