| Exam Board | Edexcel |
|---|---|
| Module | FM1 AS (Further Mechanics 1 AS) |
| Year | 2020 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Work done and energy |
| Type | Loss of energy in collision |
| Difficulty | Standard +0.8 This is a multi-part Further Mechanics collision problem requiring systematic application of conservation of momentum, Newton's experimental law (restitution), and impulse-momentum theorem. Part (a) requires careful analysis of post-collision velocities to determine conditions preventing further collisions, while part (b) involves solving simultaneous equations from the collision laws. The algebraic manipulation is substantial and requires clear physical reasoning about collision sequences. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| Scheme | Marks | Guidance |
| Use of CLM | M1 | Correct no. of terms, condone extra \(g\) \(s\), sign errors |
| \(4mu = 4mv_b + kmv_c\) | A1 | Correct equation |
| Use of NLR | M1 | e must be on correct side |
| \(\frac{1}{4}u = -v_b + v_c\) | A1 | Correct equation |
| Solve for \(v_b\) | M1 | Complete method to solve for \(v_b\) (or a multiple of \(v_b\)) |
| \(v_b = \frac{u(16-k)}{4(k+4)}\) (\(v_c = \frac{5u}{k+4}\)) | A1 | Correct expression for their \(v_b\) or a multiple of their \(v_b\) |
| Use of \(v_b \geq 0\) and solve for \(k\) | M1 | Use of appropriate inequality, allow strict inequality for method mark |
| \((0 < ) k \leq 16\) | A1 | Cao LHS not needed, but if there it must be correct. |
| Alternative for last 4 marks: | ||
| Solve for \(v_b\) in terms of \(v_c\) only | M1 | |
| \(v_b = \frac{(16-k)v_c}{20}\) | A1 | |
| Use of \(v_b \geq 0\) and \(v_c > 0\) to solve for \(k\) | M1 | |
| \((0 < ) k \leq 16\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Scheme | Marks | Guidance |
| Impulse-momentum equation | M1 | Correct no. of terms, condone sign errors, but must be subtracting momentum terms |
| \(-3mu = 4m(v_b - u)\) (\(v_b = \frac{u}{4}\)) or \(3mu = kmv_c\) | A1 | Correct equation |
| Complete method to solve for \(k\) | M1 | Eliminate and solve for \(k\) |
| \(k = 6\) | A1 |
| Scheme | Marks | Guidance |
|--------|-------|----------|
| Use of CLM | M1 | Correct no. of terms, condone extra $g$ $s$, sign errors |
| $4mu = 4mv_b + kmv_c$ | A1 | Correct equation |
| Use of NLR | M1 | e must be on correct side |
| $\frac{1}{4}u = -v_b + v_c$ | A1 | Correct equation |
| Solve for $v_b$ | M1 | Complete method to solve for $v_b$ (or a multiple of $v_b$) |
| $v_b = \frac{u(16-k)}{4(k+4)}$ ($v_c = \frac{5u}{k+4}$) | A1 | Correct expression for their $v_b$ or a multiple of their $v_b$ |
| Use of $v_b \geq 0$ and solve for $k$ | M1 | Use of appropriate inequality, allow strict inequality for method mark |
| $(0 < ) k \leq 16$ | A1 | Cao LHS not needed, but if there it must be correct. |
| **Alternative for last 4 marks:** | | |
| Solve for $v_b$ in terms of $v_c$ only | M1 | |
| $v_b = \frac{(16-k)v_c}{20}$ | A1 | |
| Use of $v_b \geq 0$ and $v_c > 0$ to solve for $k$ | M1 | |
| $(0 < ) k \leq 16$ | A1 | |
**Notes:**
- M1 for correct no. of terms, condone sign errors
**Question 3b:**
| Scheme | Marks | Guidance |
|--------|-------|----------|
| Impulse-momentum equation | M1 | Correct no. of terms, condone sign errors, but must be subtracting momentum terms |
| $-3mu = 4m(v_b - u)$ ($v_b = \frac{u}{4}$) or $3mu = kmv_c$ | A1 | Correct equation |
| Complete method to solve for $k$ | M1 | Eliminate and solve for $k$ |
| $k = 6$ | A1 | |\begin{enumerate}
\item Three particles $A , B$ and $C$ are at rest on a smooth horizontal plane. The particles lie along a straight line with $B$ between $A$ and $C$.
\end{enumerate}
Particle $B$ has mass $4 m$ and particle $C$ has mass $k m$, where $k$ is a positive constant. Particle $B$ is projected with speed $u$ along the plane towards $C$ and they collide directly.
The coefficient of restitution between $B$ and $C$ is $\frac { 1 } { 4 }$\\
(a) Find the range of values of $k$ for which there would be no further collisions.
The magnitude of the impulse on $B$ in the collision between $B$ and $C$ is $3 m u$\\
(b) Find the value of $k$.
\hfill \mbox{\textit{Edexcel FM1 AS 2020 Q3 [12]}}