Find unknown constant from motion condition

A question is this type if and only if the position or velocity expression contains an unknown constant (e.g. k or λ), and the task requires using a given condition on the direction or magnitude of motion to determine that constant.

2 questions · Standard +0.3

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OCR H240/03 2018 September Q8
8 marks Standard +0.3
8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).
  1. Show that \(k = - 5\).
  2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).
OCR H240/03 2019 June Q10
13 marks Standard +0.3
In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(R\) of mass \(2\) kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, the velocity \(\mathbf{v} \text{ m s}^{-1}\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf{v} = (pt^2 - 3t)\mathbf{i} + (8t + q)\mathbf{j}\), where \(p\) and \(q\) are constants and \(p < 0\).
  1. Given that when \(t = 0.5\) the magnitude of \(\mathbf{F}\) is \(20\), find the value of \(p\). [6]
When \(t = 0\), \(R\) is at the point with position vector \((2\mathbf{i} - 3\mathbf{j})\) m.
  1. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). [4]
When \(t = 1\), \(R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2\mathbf{i} - 8\mathbf{j}\).
  1. Find the value of \(q\). [3]