1 A ball of mass \(m\) is travelling vertically downwards with speed \(u\) when it hits a horizontal floor. The ball bounces vertically upwards to a height \(h\). It is thought that \(h\) depends on \(m , u\), the acceleration due to gravity \(g\), and a dimensionless constant \(k\), such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).

## Question 1:

| Working/Answer | Marks | Guidance |
|---|---|---|
| $L = M^\alpha(LT^{-1})^\beta(LT^{-2})^\gamma$ | M1A1 | |
| $\beta + \gamma = 1$ | | |
| $-\beta - 2\gamma = 0$ | | |
| $\alpha = 0$ | m1 | Getting three equations |
| $\gamma = -1$ | m1 | Solution |
| $\beta = 2$ | A1F | |
| **Total** | **5** | |

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1 A ball of mass $m$ is travelling vertically downwards with speed $u$ when it hits a horizontal floor. The ball bounces vertically upwards to a height $h$.

It is thought that $h$ depends on $m , u$, the acceleration due to gravity $g$, and a dimensionless constant $k$, such that

$$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$

where $\alpha , \beta$ and $\gamma$ are constants.\\
By using dimensional analysis, find the values of $\alpha , \beta$ and $\gamma$.

\hfill \mbox{\textit{AQA M3 2009 Q1 [5]}}