A particle \(P\) moves along the \(x\)-axis in the positive direction. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(\frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 6 } t } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). When \(t = 0\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Express \(v\) in terms of \(t\).
- Find, to 3 significant figures, the speed of \(P\) when \(t = 3\).
- Find the limiting value of \(v\).
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\caption{Figure 1}
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A smooth solid hemisphere, of radius 0.8 m and centre \(O\), is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the highest point \(A\) of the hemisphere. The particle leaves the hemisphere at the point \(B\), which is a vertical distance of 0.2 m below the level of \(A\). The speed of the particle at \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the angle between \(O A\) and \(O B\) is \(\theta\), as shown in Fig. 1.