| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Impulse from variable force (integration only) |
| Difficulty | Moderate -0.3 This is a straightforward M3 impulse question requiring integration of a linear force function and application of the impulse-momentum theorem. Part (a) is direct integration, part (b) requires solving a quadratic equation from the impulse-momentum relationship. Standard technique application with no novel insight needed, slightly easier than average A-level due to the simple force function. |
| Spec | 3.03c Newton's second law: F=ma one dimension6.03e Impulse: by a force6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) impulse = \(\int_1^4 F \, dr = \int_1^4 (4t+3) \, dt\) | M1 | |
| \(= [2t^2 + 3t]_1^4 = (32+12)-(2+3) = 39 \text{ Ns}\) | M1 A1 | |
| (b) \(\int_0^T F \, dr = m(v-u)\) | M1 | |
| \(\therefore [2t^2 + 3t]_0^T = 1.5(22-0)\) | A1 | |
| \(2T^2 + 3T - 33 = 0\) | A1 | |
| quad. form. gives \(T = 4.88, 3.38; T > 0 \therefore T = 3.38 \text{ s (3sf)}\) | M1 A1 | (8) |
**(a)** impulse = $\int_1^4 F \, dr = \int_1^4 (4t+3) \, dt$ | M1 |
$= [2t^2 + 3t]_1^4 = (32+12)-(2+3) = 39 \text{ Ns}$ | M1 A1 |
**(b)** $\int_0^T F \, dr = m(v-u)$ | M1 |
$\therefore [2t^2 + 3t]_0^T = 1.5(22-0)$ | A1 |
$2T^2 + 3T - 33 = 0$ | A1 |
quad. form. gives $T = 4.88, 3.38; T > 0 \therefore T = 3.38 \text{ s (3sf)}$ | M1 A1 | **(8)**
---\begin{enumerate}
\item A particle $P$ of mass 1.5 kg moves from rest at the origin such that at time $t$ seconds it is subject to a single force of magnitude $( 4 t + 3 ) \mathrm { N }$ in the direction of the positive $x$-axis.\\
(a) Find the magnitude of the impulse exerted by the force during the interval $1 \leq t \leq 4$.
\end{enumerate}
Given that at time $T$ seconds, $P$ has a speed of $22 \mathrm {~ms} ^ { - 1 }$,\\
(b) find the value of $T$ correct to 3 significant figures.\\
\hfill \mbox{\textit{Edexcel M3 Q1 [8]}}