| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Variable acceleration with initial conditions |
| Difficulty | Moderate -0.3 This is a straightforward mechanics question requiring integration of a power function with fractional exponent to find velocity, then using initial conditions. Part (i) is routine integration and substitution (answer given), part (ii) requires one more integration. The fractional power makes it slightly less routine than integer powers, but the method is standard M1 material with no problem-solving insight required. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For an attempt to find \(v(t)\) using integration of \(a(t)\) | |
| \(v = 1.2t^{5/3} + 2\) | A1 | |
| DM1 | For attempting to solve \(v(t) = 3\) for \(t^{5/3}\) or confirming \(v = 3\) by substituting \(t^{5/3} = 5/6\) into the expression found for \(v(t)\) | |
| \(t^{5/3} = 5/6\) | A1 | [4] AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For integrating and using \(s(0) = 0\) (may be implied by absence of \(+C\)) to find \(s(t)\) | |
| \(s = 0.45t^{8/3} + 2t\) | A1 | |
| Distance is 2.13 m | A1 | [3] |
# Question 3:
**(i)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For an attempt to find $v(t)$ using integration of $a(t)$ |
| $v = 1.2t^{5/3} + 2$ | A1 | |
| | DM1 | For attempting to solve $v(t) = 3$ for $t^{5/3}$ or confirming $v = 3$ by substituting $t^{5/3} = 5/6$ into the expression found for $v(t)$ |
| $t^{5/3} = 5/6$ | A1 | [4] AG |
**(ii)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For integrating and using $s(0) = 0$ (may be implied by absence of $+C$) to find $s(t)$ |
| $s = 0.45t^{8/3} + 2t$ | A1 | |
| Distance is 2.13 m | A1 | [3] |
---3 A particle $P$ moves in a straight line, starting from the point $O$ with velocity $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The acceleration of $P$ at time $t \mathrm {~s}$ after leaving $O$ is $2 t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }$.\\
(i) Show that $t ^ { \frac { 5 } { 3 } } = \frac { 5 } { 6 }$ when the velocity of $P$ is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(ii) Find the distance of $P$ from $O$ when the velocity of $P$ is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\hfill \mbox{\textit{CAIE M1 2012 Q3 [7]}}