| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Projectiles |
| Type | Projectile passing through given point |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring multiple sophisticated techniques: energy conservation to find speed at detachment, projectile trajectory equation in a non-standard coordinate system, and circular motion tension analysis. The trajectory part (finding V) requires careful geometric setup and algebraic manipulation, while part (b) needs applying circular motion principles at the initial position. The multi-stage nature and need to connect circular motion with projectile motion elevates this above standard A-level questions. |
| Spec | 3.02i Projectile motion: constant acceleration model6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | Coordinates of A: x=asin60, y=a−acos60 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | M1 | Substitute their (x, y) into correct trajectory equation. |
| Rearrange to find V2. | M1 |
| Answer | Marks |
|---|---|
| 2 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 7(b) | 1 mu2 − 1 mV2 =mga ( 1+cos60 ) | |
| 2 2 | M1 | Energy equation. |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | A1 | u is the speed at P. |
| Answer | Marks | Guidance |
|---|---|---|
| a | M1 | N2L |
| Answer | Marks |
|---|---|
| 2 | A1 |
Question 7:
--- 7(a) ---
7(a) | Coordinates of A: x=asin60, y=a−acos60 | B1
( )
2
a 3
g
a a 3 22
= 3−
2 2 1
2V2.
4 | M1 | Substitute their (x, y) into correct trajectory equation.
Rearrange to find V2. | M1
3 3
V2 = ag, V = ag
2 2 | A1
4
--- 7(b) ---
7(b) | 1 mu2 − 1 mV2 =mga ( 1+cos60 )
2 2 | M1 | Energy equation.
9
u2 = ag
2 | A1 | u is the speed at P.
m
T −mg = u2
a | M1 | N2L
11
T = mg
2 | A1
4One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $m$. The particle $P$ is held vertically below $O$ with the string taut and then projected horizontally. When the string makes an angle of $60°$ with the upward vertical, $P$ becomes detached from the string. In its subsequent motion, $P$ passes through the point $A$ which is a distance $a$ vertically above $O$.
\begin{enumerate}[label=(\alph*)]
\item The speed of $P$ when it becomes detached from the string is $V$. Use the equation of the trajectory of a projectile to find $V$ in terms of $a$ and $g$. [4]
\item Find, in terms of $m$ and $g$, the tension in the string immediately after $P$ is initially projected horizontally. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2021 Q7 [8]}}