| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2016 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Velocity from displacement differentiation |
| Difficulty | Moderate -0.8 This is a straightforward differentiation exercise requiring students to find velocity (ds/dt) and acceleration (d²s/dt²), set acceleration to zero, then substitute back. It involves routine calculus with fractional powers but no problem-solving insight or multi-step reasoning beyond standard procedure. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([v = 4t - 40t^{0.5}]\) | M1* | For differentiating \(s\) to find \(v\) |
| \([a = 4 - 20t^{-0.5}]\) | M1* | For differentiating \(v\) to find \(a\) |
| \([4 - 20t^{-0.5} = 0]\) | DM1 | For setting \(a = 0\) and attempt to solve to find \(t\) |
| \(t = 25 \text{ s}\) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitute their \(t\) into \(s\) or \(v\) | M1 | |
| Displacement \(= -2083.3 \text{ m}\) \((= -2080 \text{ 3sf})\) and Velocity \(= -100 \text{ ms}^{-1}\) | A1 [2] | or Displacement \(= -6250/3\) |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[v = 4t - 40t^{0.5}]$ | M1* | For differentiating $s$ to find $v$ |
| $[a = 4 - 20t^{-0.5}]$ | M1* | For differentiating $v$ to find $a$ |
| $[4 - 20t^{-0.5} = 0]$ | DM1 | For setting $a = 0$ and attempt to solve to find $t$ |
| $t = 25 \text{ s}$ | A1 [4] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute their $t$ into $s$ or $v$ | M1 | |
| Displacement $= -2083.3 \text{ m}$ $(= -2080 \text{ 3sf})$ **and** Velocity $= -100 \text{ ms}^{-1}$ | A1 [2] | or Displacement $= -6250/3$ |
---2 A particle moves in a straight line. Its displacement $t \mathrm {~s}$ after leaving a fixed point $O$ on the line is $s \mathrm {~m}$, where $s = 2 t ^ { 2 } - \frac { 80 } { 3 } t ^ { \frac { 3 } { 2 } }$.\\
(i) Find the time at which the acceleration of the particle is zero.\\
(ii) Find the displacement and velocity of the particle at this instant.
\hfill \mbox{\textit{CAIE M1 2016 Q2 [6]}}