A ball of mass 0.4 kg is thrown vertically upwards from a point \(O\) with initial speed \(17 \mathrm {~ms} ^ { - 1 }\). When the ball is at a height of \(x \mathrm {~m}\) above \(O\) and its speed is \(v \mathrm {~ms} ^ { - 1 }\), the air resistance acting on the ball has magnitude \(0.01 v ^ { 2 } \mathrm {~N}\).
Show that, as the ball is ascending, \(v\) satisfies the differential equation
$$40 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 392 + v ^ { 2 } \right)$$
Find an expression for \(v\) in terms of \(x\).
Calculate, correct to two decimal places, the greatest height of the ball.
State, with a reason, whether the speed of the ball when it returns to \(O\) is greater than \(17 \mathrm {~ms} ^ { - 1 }\), less than \(17 \mathrm {~ms} ^ { - 1 }\) or equal to \(17 \mathrm {~ms} ^ { - 1 }\).