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.
[0pt]
[1 mark]
| 19 | | 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. | | [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) |
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