| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Constant acceleration (SUVAT) |
| Type | Sketch velocity-time graph |
| Difficulty | Moderate -0.3 This is a straightforward mechanics question requiring standard calculus techniques (integration to find distance, differentiation to find maximum velocity) applied to piecewise velocity functions. While it has multiple parts and requires careful attention to sign conventions when the particle reverses direction, all techniques are routine for M1 level with no novel problem-solving required. |
| Spec | 1.07a Derivative as gradient: of tangent to curve3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Velocity immediately before is \(1.2\ \text{ms}^{-1}\) | B1 | |
| Velocity immediately after is \(-1\ \text{ms}^{-1}\) | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For using distance \(OW = \int v\, dt\) with limits \(0\) to \(60\) (W is wall) or for using distance \(WA = -\int v\, dt\) with limits \(60\) to \(100\) | |
| Distance \(OW = 0.025 \times 60^2 - 0.0005 \times 60^3 \div 3\) | A1 | |
| Distance \(WA = -[(0.0125 \times 100^2 - 2.5 \times 100) - (0.0125 \times 60^2 - 2.5 \times 60)]\) | A1 | |
| Distance is \(54 + 20 = 74\) m | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dv}{dt} = 0.05 - 0.001t = 0\) or \(0.0005t(100-t) = 0 \rightarrow t = 0\) or \(100\) | M1 | For using \(v_{max}\) occurs when \(\frac{dv}{dt} = 0\) or when \(t\) = the midpoint of the roots of the quadratic equation \(v = 0\) |
| Maximum speed \(= 0.05 \times 50 - 0.0005 \times 50^2\) is \(1.25\text{ ms}^{-1}\) | A1 | |
| Plausible quadratic curve starting at \((0,0)\), with max. at \((50, 1.25)\) and terminating at \((60, 1.2)\) | B1 | |
| Straight line segment from \((60,-1)\) to \((100,0)\) | B1 | [4] |
## Question 6:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Velocity immediately before is $1.2\ \text{ms}^{-1}$ | B1 | |
| Velocity immediately after is $-1\ \text{ms}^{-1}$ | B1 [2] | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For using distance $OW = \int v\, dt$ with limits $0$ to $60$ (W is wall) or for using distance $WA = -\int v\, dt$ with limits $60$ to $100$ |
| Distance $OW = 0.025 \times 60^2 - 0.0005 \times 60^3 \div 3$ | A1 | |
| Distance $WA = -[(0.0125 \times 100^2 - 2.5 \times 100) - (0.0125 \times 60^2 - 2.5 \times 60)]$ | A1 | |
| Distance is $54 + 20 = 74$ m | A1 [4] | |
## Question 6(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dv}{dt} = 0.05 - 0.001t = 0$ or $0.0005t(100-t) = 0 \rightarrow t = 0$ or $100$ | M1 | For using $v_{max}$ occurs when $\frac{dv}{dt} = 0$ or when $t$ = the midpoint of the roots of the quadratic equation $v = 0$ |
| Maximum speed $= 0.05 \times 50 - 0.0005 \times 50^2$ is $1.25\text{ ms}^{-1}$ | A1 | |
| Plausible quadratic curve starting at $(0,0)$, with max. at $(50, 1.25)$ and terminating at $(60, 1.2)$ | B1 | |
| Straight line segment from $(60,-1)$ to $(100,0)$ | B1 | [4] |
---6 A particle starts from rest at a point $O$ and moves in a horizontal straight line. The velocity of the particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$ after leaving $O$. For $0 \leqslant t < 60$, the velocity is given by
$$v = 0.05 t - 0.0005 t ^ { 2 }$$
The particle hits a wall at the instant when $t = 60$, and reverses the direction of its motion. The particle subsequently comes to rest at the point $A$ when $t = 100$, and for $60 < t \leqslant 100$ the velocity is given by
$$v = 0.025 t - 2.5$$
(i) Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.\\
(ii) Find the total distance travelled by the particle.\\
(iii) Find the maximum speed of the particle and sketch the particle's velocity-time graph for $0 \leqslant t \leqslant 100$, showing the value of $t$ for which the speed is greatest.
\section*{[Question 7 is printed on the next page.]}
\hfill \mbox{\textit{CAIE M1 2014 Q6 [10]}}