15 A model for the motion of a small object falling through a thick fluid can be expressed using the differential equation
\(\frac { \mathrm { d } v } { \mathrm {~d} t } = 9.8 - k v\),
where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity after \(t \mathrm {~s}\) and \(k\) is a positive constant.
- Given that \(v = 0\) when \(t = 0\), solve the differential equation to find \(v\) in terms of \(t\) and \(k\).
- Sketch the graph of \(v\) against \(t\).
Experiments show that for large values of \(t\), the velocity tends to \(7 \mathrm {~ms} ^ { - 1 }\).
- Find the value of \(k\).
- Find the value of \(t\) for which \(v = 3.5\).