| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Variable acceleration with initial conditions |
| Difficulty | Standard +0.3 This is a straightforward integration problem with initial conditions. Students integrate the given acceleration twice (using standard power rule after rewriting), apply initial conditions to find constants, then substitute t=3. The algebra is clean and the integration is routine for M3 level, making this slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(v = \int \frac{4}{(1+t)^3} \text{ d}t\) \(\therefore v = \frac{-2}{(1+t)^2} + c\) | M1 A1 | |
| t = 0, v = 1 \(\therefore c = 3\) giving \(v = \left[3 - \frac{2}{(1+t)^2}\right] \text{ m s}^{-1}\) | M1; A1 | |
| (b) \(x = \int 3 - \frac{2}{(1+t)^2} \text{ d}t\) \(\therefore x = 3t + \frac{1}{(1+t)} + d\) | M1 A1 | |
| t = 0, x = 3 \(\therefore d = 1\) giving \(x = 3t + \frac{1}{(1+t)} + 1\) | M1 | |
| t = 3, x = 9 + \(\frac{1}{4}\) + 1 = 10.5 m | M1 A1 | (9) |
(a) $v = \int \frac{4}{(1+t)^3} \text{ d}t$ $\therefore v = \frac{-2}{(1+t)^2} + c$ | M1 A1 |
t = 0, v = 1 $\therefore c = 3$ giving $v = \left[3 - \frac{2}{(1+t)^2}\right] \text{ m s}^{-1}$ | M1; A1 |
(b) $x = \int 3 - \frac{2}{(1+t)^2} \text{ d}t$ $\therefore x = 3t + \frac{1}{(1+t)} + d$ | M1 A1 |
t = 0, x = 3 $\therefore d = 1$ giving $x = 3t + \frac{1}{(1+t)} + 1$ | M1 |
t = 3, x = 9 + $\frac{1}{4}$ + 1 = 10.5 m | M1 A1 | (9)
---3. At time $t$ seconds the acceleration, $a \mathrm {~ms} ^ { - 2 }$, of a particle is given by
$$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$
When $t = 0$, the particle has velocity $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and displacement 3 m from a fixed origin $O$.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for the velocity of the particle in terms of $t$.
\item Show that when $t = 3$ the particle is 10.5 m from $O$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q3 [9]}}