| Exam Board | OCR |
|---|---|
| Module | FM1 AS (Further Mechanics 1 AS) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Dimensional Analysis |
| Type | Find exponents with partial constraints |
| Difficulty | Standard +0.3 This is a structured dimensional analysis question with clear scaffolding through parts (a)-(d). While it requires understanding of dimensions and the suvat equations, each part guides students through the process systematically. The final part requires recognizing the work-energy theorem, but this is standard FM1 content. Slightly easier than average due to the step-by-step guidance. |
| Spec | 3.02d Constant acceleration: SUVAT formulae6.01a Dimensions: M, L, T notation6.01b Units vs dimensions: relationship6.01c Dimensional analysis: error checking6.01d Unknown indices: using dimensions |
A particle of mass $m$ moves in a straight line with constant acceleration $a$. Its initial and final velocities are $u$ and $v$ respectively and its final displacement from its starting position is $s$. In order to model the motion of the particle it is suggested that the velocity is given by the equation
$$v^2 = pu^{\alpha} + qa^{\beta}s^{\gamma}$$
where $p$ and $q$ are dimensionless constants.
\begin{enumerate}[label=(\alph*)]
\item Explain why $\alpha$ must equal 2 for the equation to be dimensionally consistent. [2]
\item By using dimensional analysis, determine the values of $\beta$ and $\gamma$. [4]
\item By considering the case where $s = 0$, determine the value of $p$. [1]
\item By multiplying both sides of the equation by $\frac{1}{2}m$, and using the numerical values of $\alpha$, $\beta$ and $\gamma$, determine the value of $q$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FM1 AS 2021 Q3 [9]}}