2 A pile driver of mass \(m _ { 1 }\) falls from a height \(h\) onto a pile of mass \(m _ { 2 }\), driving the pile a distance \(s\) into the ground. The pile driver remains in contact with the pile after the impact. A resistance force \(R\) opposes the motion of the pile into the ground. Elizabeth finds an expression for \(R\) as $$R = \frac { g } { s } \left[ s \left( m _ { 1 } + m _ { 2 } \right) + \frac { h \left( m _ { 1 } \right) ^ { 2 } } { m _ { 1 } + m _ { 2 } } \right]$$ where \(g\) is the acceleration due to gravity.
Determine whether the expression is dimensionally consistent.

# Question 2:

| Working/Answer | Mark | Guidance |
|---|---|---|
| $[R] = $ MLT$^{-2}$ (dimensions of force) | B1 | Stating dimensions of $R$ |
| $\left[\frac{g}{s}\right] = \frac{\text{LT}^{-2}}{\text{L}} = \text{T}^{-2}$ | M1 | Finding dimensions of $\frac{g}{s}$ |
| $[s(m_1+m_2)] = \text{L} \cdot \text{M} = \text{ML}$ | A1 | Correct dimensions of first bracket term |
| $\left[\frac{h(m_1)^2}{m_1+m_2}\right] = \frac{\text{L} \cdot \text{M}^2}{\text{M}} = \text{ML}$ | A1 | Correct dimensions of second term, concluding both terms same, so expression dimensionally consistent giving MLT$^{-2}$ |

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2 A pile driver of mass $m _ { 1 }$ falls from a height $h$ onto a pile of mass $m _ { 2 }$, driving the pile a distance $s$ into the ground. The pile driver remains in contact with the pile after the impact. A resistance force $R$ opposes the motion of the pile into the ground.

Elizabeth finds an expression for $R$ as

$$R = \frac { g } { s } \left[ s \left( m _ { 1 } + m _ { 2 } \right) + \frac { h \left( m _ { 1 } \right) ^ { 2 } } { m _ { 1 } + m _ { 2 } } \right]$$

where $g$ is the acceleration due to gravity.\\
Determine whether the expression is dimensionally consistent.

\hfill \mbox{\textit{AQA M3 2012 Q2 [4]}}