| Exam Board | CAIE |
|---|---|
| Module | Further Paper 3 (Further Paper 3) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Oblique collision of spheres |
| Difficulty | Challenging +1.8 This is an advanced mechanics problem requiring conservation of momentum in 2D, Newton's experimental law with e=1/2, and the geometric constraint that both spheres reach walls simultaneously. It demands careful vector decomposition, multiple equations, and algebraic manipulation beyond standard A-level, but follows systematic collision mechanics principles without requiring exceptional insight. |
| Spec | 6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks |
|---|---|
| 4(a) | After collision, A has velocity v towards wall on right and B has component |
| Answer | Marks |
|---|---|
| A | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| A B | M1 | Both, consistent signs, must be cos. |
| 2sin=(1+e)cos | M1 | Eliminating v and v to find an equation in . |
| Answer | Marks |
|---|---|
| 4 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 4(b) | 1 ( ) |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | B1 | Both components of velocity of B needed. |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 2 25 25 25 | M1 | v , v , substituted, ft their final KE with both |
| Answer | Marks |
|---|---|
| Percentage loss = 24% | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(a) ---
4(a) | After collision, A has velocity v towards wall on right and B has component
A
of velocity towards lower wall of u sin . Same distance and time, so
v = u sin .
A | B1
Along line of centres:
PCLM: mv +mv =mucos
A B
NEL: v −v =eucos
A B | M1 | Both, consistent signs, must be cos.
2sin=(1+e)cos | M1 | Eliminating v and v to find an equation in .
A B
Condone common factor of u.
3 1
Note: v = ucos and v = ucos.
A B
4 4
tan= 3
4 | A1
4
Question | Answer | Marks | Guidance
--- 4(b) ---
4(b) | 1 ( )
Final KE = m v2 +(usin)2 +v2
A B
2 | B1 | Both components of velocity of B needed.
1 1 ( ) 1 9 9 1
Loss = mu2 − m v2 +(usin)2 +v2 = mu2 1− − −
A B
2 2 2 25 25 25 | M1 | v , v , substituted, ft their final KE with both
A B
components of velocity of B included.
3
Loss = mu2
25
Percentage loss = 24% | A1
3
Question | Answer | Marks | Guidance\includegraphics{figure_4}
Two smooth vertical walls meet at right angles. The smooth sphere $A$, with mass $m$, is at rest on a smooth horizontal surface and is at a distance $d$ from each wall. An identical smooth sphere $B$ is moving on the horizontal surface with speed $u$ at an angle $\theta$ with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is $\frac{1}{2}$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\tan \theta$. [4]
\item Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 3 2023 Q4 [7]}}