| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Variable acceleration with initial conditions |
| Difficulty | Standard +0.3 This is a standard kinematics question requiring double integration of acceleration with boundary conditions, finding stationary points, and calculating total distance. While it involves multiple steps (integration, solving simultaneous equations, finding roots, evaluating displacement at critical points), all techniques are routine M1 procedures with no novel insight required. Slightly easier than average due to straightforward polynomial integration and simple arithmetic. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([v = 6t^2/2 - 12t + C]\) \(\quad v = 3t^2 - 12t + C\) | *M1 | Use of \(v = \int a \, dt\) |
| \([s = 3t^3/3 - 12t^2/2 + Ct + D]\) \(\quad s = t^3 - 6t^2 + Ct + D\) | *M1 | Use of \(s = \int v \, dt\) |
| \(5 = 1 - 6 + C + D \quad C + D = 10\) and \(1 = 27 - 54 + 3C + D \quad 3C + D = 28 \rightarrow C = ....., D = .....\) | DM1 | Substitutes for \(s\) and \(t\) and solves equations. Dependent on both Ms. |
| \(s = t^3 - 6t^2 + 9t + 1\) or \(p = 9, q = 1\) | A1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([v = 0, 3t^2 - 12t + 9 = 0 \Rightarrow (t-1)(t-3) = 0 \rightarrow t = .....]\) | M1 | Solves \(v = 0\) to find \(t\) values |
| \(t = 1\) or \(t = 3\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[\int_0^1 v\,dt + \int_1^3 v\,dt + \int_3^4 v\,dt\right]\) | M1 | Attempts to use at least three \(t\) intervals |
| [For \(0 \leq t \leq 1\), \(s = (1-6+9+1)-1 = 4\)] | M1 | Evaluates \(s\) for one time interval |
| \(0 \leq t \leq 1\), \(s = (1-6+9+1)-1 = \mathbf{4}\); \(1 \leq t \leq 3\), \(s = (27-54+27+1)-5 = \mathbf{-4}\); \(3 \leq t \leq 4\), \(s = (64-96+36+1)-1 = \mathbf{4}\) | A1 | Correctly finds all at least two distances (ignoring signs) |
| Total distance is \(12\text{ m}\) | A1 | |
| 4 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[v = 6t^2/2 - 12t + C]$ $\quad v = 3t^2 - 12t + C$ | *M1 | Use of $v = \int a \, dt$ |
| $[s = 3t^3/3 - 12t^2/2 + Ct + D]$ $\quad s = t^3 - 6t^2 + Ct + D$ | *M1 | Use of $s = \int v \, dt$ |
| $5 = 1 - 6 + C + D \quad C + D = 10$ and $1 = 27 - 54 + 3C + D \quad 3C + D = 28 \rightarrow C = ....., D = .....$ | DM1 | Substitutes for $s$ and $t$ and solves equations. Dependent on both Ms. |
| $s = t^3 - 6t^2 + 9t + 1$ or $p = 9, q = 1$ | A1 | |
| | **4** | |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[v = 0, 3t^2 - 12t + 9 = 0 \Rightarrow (t-1)(t-3) = 0 \rightarrow t = .....]$ | M1 | Solves $v = 0$ to find $t$ values |
| $t = 1$ or $t = 3$ | A1 | |
| | **2** | |
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\int_0^1 v\,dt + \int_1^3 v\,dt + \int_3^4 v\,dt\right]$ | M1 | Attempts to use at least three $t$ intervals |
| [For $0 \leq t \leq 1$, $s = (1-6+9+1)-1 = 4$] | M1 | Evaluates $s$ for one time interval |
| $0 \leq t \leq 1$, $s = (1-6+9+1)-1 = \mathbf{4}$; $1 \leq t \leq 3$, $s = (27-54+27+1)-5 = \mathbf{-4}$; $3 \leq t \leq 4$, $s = (64-96+36+1)-1 = \mathbf{4}$ | A1 | Correctly finds all at least two distances (ignoring signs) |
| Total distance is $12\text{ m}$ | A1 | |
| | **4** | |6 A particle $P$ moves in a straight line. The acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ of $P$ at time $t \mathrm {~s}$ is given by $a = 6 t - 12$. The displacement of $P$ from a fixed point $O$ on the line is $s \mathrm {~m}$. It is given that $s = 5$ when $t = 1$ and $s = 1$ when $t = 3$.\\
(i) Show that $s = t ^ { 3 } - 6 t ^ { 2 } + p t + q$, where $p$ and $q$ are constants to be found.\\
(ii) Find the values of $t$ when $P$ is at instantaneous rest.\\
(iii) Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 4$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE M1 2019 Q6 [10]}}