| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Velocity from acceleration by integration |
| Difficulty | Moderate -0.5 This is a straightforward variable acceleration problem requiring integration of a given acceleration function (though the function appears corrupted in the question text), applying initial conditions, and integrating velocity to find displacement. These are standard M2 techniques with no novel problem-solving required, making it slightly easier than average but still requiring multiple calculus steps. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| 2 (i) | 0.5vdv/dx = 0.5g – 0.015x2 | |
| vdv/dx = 10 – 0.03x2 AG | M1 | |
| A1 | [2] | N2L 2 forces |
| (ii) | 18.3 m | B1 |
| Answer | Marks |
|---|---|
| (iii) | ∫ vdv = ∫ (10–0.03x2 )dx |
| Answer | Marks |
|---|---|
| v = 15.6 ms −1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| A1 | [4] | Attempts to integrate |
| Answer | Marks |
|---|---|
| Uses ans (ii) in formula for | 7 |
Question 2:
--- 2 (i) ---
2 (i) | 0.5vdv/dx = 0.5g – 0.015x2
vdv/dx = 10 – 0.03x2 AG | M1
A1 | [2] | N2L 2 forces
(ii) | 18.3 m | B1 | [1] | ( )
10/0.03 =18.257..
(iii) | ∫ vdv = ∫ (10–0.03x2 )dx
v2 /2 = 10x – 0.03x3 /3 (+c)
v2 /2 = 10× 18.3 – 0.03×18.3 3 /3
v = 15.6 ms −1 | M1
A1
M1
A1 | [4] | Attempts to integrate
Accept omission of c
v2
Uses ans (ii) in formula for | 7A particle moves in a straight line. At time $t$ seconds its velocity is $v$ ms$^{-1}$ and its acceleration is $a$ ms$^{-2}$.
\begin{enumerate}[label=(\roman*)]
\item Given that $a = —$, express $v$ in terms of $t$.
\item Given that $v = tv$ when $t = 0$, find $v$ in terms of $t$.
\item Find the displacement from the starting point when $t = v$.
\end{enumerate}
[6]
\hfill \mbox{\textit{CAIE M2 2013 Q2 [6]}}