| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2020 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string with compression (spring) |
| Difficulty | Challenging +1.2 This is a multi-step mechanics problem combining conservation of momentum (for the collision), energy conservation (for the spring motion), and Hooke's law. While it requires careful bookkeeping of energy terms (kinetic + elastic) and involves algebraic manipulation with parameters, the techniques are standard for Further Maths mechanics. The problem is structured with clear stages (collision → subsequent motion) and doesn't require novel insight beyond applying familiar principles systematically. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.03b Conservation of momentum: 1D two particles |
| Answer | Marks |
|---|---|
| 7(a) | Collision: ( k+1 ) mv=mu so v= u |
| k+1 | B1 |
| Answer | Marks |
|---|---|
| 2 4 | B1 |
| Answer | Marks |
|---|---|
| 4 | B1 |
| Answer | Marks |
|---|---|
| 2 4 ( k+1 )2 8 | M1 |
| Use value of u and solve | M1 |
| k =3 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 7(b) | 9g |
| Answer | Marks |
|---|---|
| 20 | M1 |
| Answer | Marks |
|---|---|
| a 5 | M1 |
| μR=4mg/5and compare with R=4mg | B1 |
| Answer | Marks |
|---|---|
| 5 | A1 |
Question 7:
--- 7(a) ---
7(a) | Collision: ( k+1 ) mv=mu so v= u
k+1 | B1
Loss in KE = 1 m ( k+1 ) v2 − 1 v2
2 4 | B1
2
a
Gain in EPE = ½ .4mg/a.
4 | B1
So, 1 . 3 .m ( k+1 ) u2 = mga
2 4 ( k+1 )2 8 | M1
Use value of u and solve | M1
k =3 | A1
6
Question | Answer | Marks
--- 7(b) ---
7(b) | 9g
T +F =4m.
20 | M1
a
9mg
4mg.4 +F =
a 5 | M1
μR=4mg/5and compare with R=4mg | B1
1
μ=
5 | A1
4\includegraphics{figure_7}
One end of a light spring of natural length $a$ and modulus of elasticity $4mg$ is attached to a fixed point $O$. The other end of the spring is attached to a particle $A$ of mass $km$, where $k$ is a constant. Initially the spring lies at rest on a smooth horizontal surface and has length $a$. A second particle $B$, of mass $m$, is moving towards $A$ with speed $\sqrt{\frac{4}{3}ga}$ along the line of the spring from the opposite direction to $O$ (see diagram).
The particles $A$ and $B$ collide and coalesce. At a point $C$ in the subsequent motion, the length of the spring is $\frac{5}{4}a$ and the speed of the combined particle is half of its initial speed.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $k$. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q7 [6]}}