A particle is travelling in a straight line.
It accelerates from its initial velocity \(u\) to
velocity \(v\), where \(v > \vert u \vert > 0\,\), travelling a distance \(d_1\)
with uniform acceleration of magnitude \(3a\,\).
It then comes to rest after travelling
a further distance \(d_2\,\) with uniform deceleration of magnitude \(a\,\).
Show that
- if \(u>0\) then \(3d_1 < d_2\,\);
- if \(u<0\) then \(d_2 < 3d_1 < 2d_2\,\).
Show also that
the average speed of the particle (that is, the total distance
travelled divided by the total time) is greater in the case \(u>0\) than in the case \(u<0\,\).
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{\bf Note:} In this question \(d_1\) and \(d_2\) are distances travelled by the particle which
are not the same, in the second case, as displacements from the starting point.