- The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula
$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$
where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
- Find the acceleration of the toy in terms of \(t\).
- Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
- Explain why this model is unlikely to be realistic for large values of \(t\).