| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Velocity from displacement differentiation |
| Difficulty | Moderate -0.3 This is a straightforward differentiation exercise requiring v = ds/dt and a = dv/dt with exponential functions, plus evaluation at boundary/initial conditions. The calculus is routine (exponential differentiation is standard at this level), and part (d) requires only basic physical reasoning about airbag behavior. Slightly easier than average due to the mechanical nature of the steps. |
| Spec | 1.06b Gradient of e^(kx): derivative and exponential model1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(s = e^{0.3} - 1 = 0.3499 m = 35\) cm (nearest cm) | M1 A1 | |
| \(v = \frac{ds}{dt} = 3e^{3t}\) | M1 A1 | |
| when \(t = 0, v = 3\) ms\(^{-1}\) | A1 | |
| \(a = \frac{dv}{dt} = 9e^{3t}\) ms\(^{-2}\) | A1 | |
| e.g. model predicts increasing accel. more likely to be decreasing | B1 | (7) |
| $s = e^{0.3} - 1 = 0.3499 m = 35$ cm (nearest cm) | M1 A1 | |
| $v = \frac{ds}{dt} = 3e^{3t}$ | M1 A1 | |
| when $t = 0, v = 3$ ms$^{-1}$ | A1 | |
| $a = \frac{dv}{dt} = 9e^{3t}$ ms$^{-2}$ | A1 | |
| e.g. model predicts increasing accel. more likely to be decreasing | B1 | (7) |\begin{enumerate}
\item A student is attempting to model the expansion of an airbag in a car following a collision.
\end{enumerate}
The student considers the displacement from the steering column, $s$ metres, of a point $P$ on the airbag $t$ seconds after a collision and uses the formula
$$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$
Using this model,\\
(a) find, correct to the nearest centimetre, the maximum displacement of $P$,\\
(b) find the initial velocity of $P$,\\
(c) find the acceleration of $P$ in terms of $t$.\\
(d) Explain why this model is unlikely to be realistic.\\
\hfill \mbox{\textit{Edexcel M3 Q1 [7]}}