18 Two particles, \(P\) and \(Q\), are projected at the same time from a fixed point \(X\), on the ground, so that they travel in the same vertical plane.
\(P\) is projected at an acute angle \(\theta ^ { \circ }\) to the horizontal, with speed \(u \mathrm {~ms} ^ { - 1 }\)
\(Q\) is projected at an acute angle \(2 \theta ^ { \circ }\) to the horizontal, with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Both particles land back on the ground at exactly the same point, \(Y\).
Resistance forces to motion may be ignored.
18
- Show that
$$\cos 2 \theta = \frac { 1 } { 8 }$$
18
- \(\quad P\) takes a total of 0.4 seconds to travel from \(X\) to \(Y\).
Find the time taken by \(Q\) to travel from \(X\) to \(Y\).
18 - State one modelling assumption you have chosen to make in this question.
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Two skaters, Jo and Amba, are separately skating across a smooth, horizontal surface of ice.
Both are moving in the same direction, so that their paths are straight and are parallel to each other.
Jo is moving with constant velocity \(( 2.8 \mathbf { i } + 9.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
At time \(t = 0\) seconds Amba is at position ( \(2 \mathbf { i } - 7 \mathbf { j }\) ) metres and is moving with a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Explain why Amba's velocity must be in the form \(k ( 2.8 \mathbf { i } + 9.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
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