| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration from velocity differentiation |
| Difficulty | Standard +0.3 This is a straightforward M3 question requiring differentiation of exponential and polynomial functions to find acceleration, then solving when acceleration is parallel to a given vector (ratio of components). Part (c) is basic interpretation. Slightly above average due to vector context and the parallelism condition, but all techniques are routine for M3 students. |
| Spec | 1.06b Gradient of e^(kx): derivative and exponential model1.10h Vectors in kinematics: uniform acceleration in vector form3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = \frac{d}{dt}(v) = (2c^2 i + 2j) \text{ cm s}^{-2}\) | M1 A1 | |
| \(2c^2 i + 2j = k(4i + j)\), comparing \(j\) components, \(k = 2\) | M1 A1 | |
| \(\therefore 2c^2 = 8, \quad t = \frac{1}{2} \ln 4 = 0.69 \text{ s (2sf)}\) | M1 A1 | |
| e.g. predicts \(v\), \(a\) increasing to very large values | B1 | (7) |
$a = \frac{d}{dt}(v) = (2c^2 i + 2j) \text{ cm s}^{-2}$ | M1 A1 |
$2c^2 i + 2j = k(4i + j)$, comparing $j$ components, $k = 2$ | M1 A1 |
$\therefore 2c^2 = 8, \quad t = \frac{1}{2} \ln 4 = 0.69 \text{ s (2sf)}$ | M1 A1 |
e.g. predicts $v$, $a$ increasing to very large values | B1 | (7)\begin{enumerate}
\item The velocity, $\mathbf { v ~ c m ~ s } { } ^ { - 1 }$, at time $t$ seconds, of a radio-controlled toy is modelled by the formula
\end{enumerate}
$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$
where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors.\\
(a) Find the acceleration of the toy in terms of $t$.\\
(b) Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector $( 4 \mathbf { i } + \mathbf { j } )$.\\
(c) Explain why this model is unlikely to be realistic for large values of $t$.\\
\hfill \mbox{\textit{Edexcel M3 Q1 [7]}}