2 George is investigating the time it takes for a ball to reach a certain height when projected vertically upwards. George believes that the time, \(t\), for the ball to reach a certain height, \(h\), depends on

• the ball's mass \(m\),
• the projection speed \(u\), and
• the height \(h\).

George suggests the following formula to model this situation
\(t = k m ^ { \alpha } u ^ { \beta } h ^ { \gamma }\),
where \(k\) is a dimensionless constant.

1. Use dimensional analysis to show that \(t = \frac { k h } { u }\).
2. Hence explain why George’s formula is unrealistic. Mandy argues that any model of this situation must consider the acceleration due to gravity, \(g\). She suggests the alternative formula
   \(t = \frac { u - \sqrt { u ^ { 2 } + g h } } { g }\).
3. Show that Mandy's formula is dimensionally consistent.
4. Explain why Mandy’s formula is incorrect.