Distance between two positions

1. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where

$$\mathbf { r } = \left( \frac { 1 } { 8 } t ^ { 4 } - 2 \lambda t ^ { 2 } + 5 \right) \mathbf { i } + \left( 5 t ^ { 2 } - \lambda t \right) \mathbf { j }$$ and \(\lambda\) is a constant. When \(t = 4 , P\) is moving parallel to the vector \(\mathbf { j }\).

1. Show that \(\lambda = 2\)
2. Find the speed of \(P\) when \(t = 4\)
3. Find the acceleration of \(P\) when \(t = 4\) When \(t = 0 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
4. Find the distance \(A B\).

4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j } .$$ When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\).

1. Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
2. Calculate the distance \(O S\).

13 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the \(x\) - and \(y\)-directions respectively.
The velocity of a particle at time \(t \mathrm {~s}\) is given by \(\left( 3 t ^ { 2 } \mathbf { i } + 7 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0\) the position of the particle with respect to the origin is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\).

1. Determine the distance of the particle from the origin when \(t = 2\).
2. Show that the cartesian equation of the path of the particle is \(x = \left( \frac { y - 2 } { 7 } \right) ^ { 3 } - 1\).
3. At time \(t = 2\), the magnitude of the resultant force acting on the particle is 48 N . Find the mass of the particle.