3 Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\).
Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her.
The time, \(t \mathrm {~s}\), is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O .
Nina's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), is given by
$$\begin{array} { l l }
a = 4 - t & \text { for } 0 \leqslant t \leqslant 4 ,
a = 0 & \text { for } t > 4 .
\end{array}$$
- Show that Nina's speed, \(v \mathrm {~ms} ^ { - 1 }\), is given by
$$\begin{array} { l l }
v = 4 t - \frac { 1 } { 2 } t ^ { 2 } & \text { for } 0 \leqslant t \leqslant 4 ,
v = 8 & \text { for } t > 4 .
\end{array}$$ - Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t \leqslant 4\).
Find how far Nina has run when \(t = 4\) and when \(t = 5 \frac { 1 } { 3 }\).
- Show that Nina catches up with Marie when \(t = 5 \frac { 1 } { 3 }\).