1 The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a wave travelling along the surface of a sea is believed to depend on
the depth of the sea, \(d \mathrm {~m}\),
the density of the water, \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\),
the acceleration due to gravity, \(g\), and
a dimensionless constant, \(k\) so that $$v = k d ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, show that \(\beta = 0\) and find the values of \(\alpha\) and \(\gamma\).

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $LT^{-1} = L^{\alpha} \times (ML^{-3})^{\beta}(LT^{-2})^{\gamma}$ | M1 | |
| There is no $M$ on the left hand side, so $\beta = 0$ | E1 | |
| $LT^{-1} = L^{\alpha+\gamma}T^{-2\gamma}$ | m1 | Dependent on M1 |
| $\alpha + \gamma = 1$ | m1 | Equating corresponding indices |
| $-2\gamma = -1$ | m1 | Equating corresponding indices |
| $\gamma = \frac{1}{2}$ | A1 | |
| $\alpha = \frac{1}{2}$ | A1 | **Total: 6** |

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1 The speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of a wave travelling along the surface of a sea is believed to depend on\\
the depth of the sea, $d \mathrm {~m}$,\\
the density of the water, $\rho \mathrm { kg } \mathrm { m } ^ { - 3 }$,\\
the acceleration due to gravity, $g$, and\\
a dimensionless constant, $k$\\
so that

$$v = k d ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }$$

where $\alpha , \beta$ and $\gamma$ are constants.\\
By using dimensional analysis, show that $\beta = 0$ and find the values of $\alpha$ and $\gamma$.

\hfill \mbox{\textit{AQA M3 2008 Q1 [6]}}