| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2020 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions |
| Type | Collision with two possible outcomes |
| Difficulty | Standard +0.3 This is a standard M1 collision problem requiring conservation of momentum and impulse-momentum theorem. Part (a) is direct application of impulse formula, part (b) uses momentum conservation with given information, and part (c) requires recognizing that speeds must be positive. The multi-step nature and the inequality derivation elevate it slightly above average, but it follows a well-practiced template with no novel insight required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\pm m_2\left(\frac{1}{3}u - -u\right)\) | M1 A1 | M1 for impulse-momentum principle applied to \(Q\); condone sign errors but must use \(m_2\) for mass and subtracting momenta. M0 if dimensionally incorrect e.g. if \(g\) included |
| \(\frac{4m_2u}{3}\) | A1 (3) | Must be positive and a single term. Allow fraction replaced by decimal to at least 2 SF |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| CLM: \(m_1u - m_2u = -m_1v + m_2\frac{1}{3}u\) OR \(\frac{4m_2u}{3} = m_1(v - -u)\) | M1 A1 | M1 CLM with usual rules (allow consistent extra \(g\)'s), or impulse-momentum principle applied to \(P\) using answer from (a). Must be in terms of \(m_2\) and \(u\) |
| \(\frac{u(4m_2 - 3m_1)}{3m_1}\) oe | A1 (3) | Correct answer only. Any equivalent expression with \(m_2\) terms collected. Allow fraction replaced by decimal to at least 2 SF. Allow consistent use of \(-v\) instead of \(v\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{u(4m_2 - 3m_1)}{3m_1} > 0\) | M1 | Correct inequality for their \(v\), containing \(u\). First statement must include \(u\) and \(> 0\) or \(< 0\) as appropriate |
| \((4m_2 - 3m_1) > 0 \Rightarrow 4m_2 > 3m_1 \Rightarrow m_2 > \frac{3}{4}m_1\) | A1* (2) | Correct given answer correctly obtained. N.B. \(\frac{3}{4}m_1 < m_2\) is A0 |
## Question 1:
### Part (a)
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\pm m_2\left(\frac{1}{3}u - -u\right)$ | M1 A1 | M1 for impulse-momentum principle applied to $Q$; condone sign errors but must use $m_2$ for mass and subtracting momenta. M0 if dimensionally incorrect e.g. if $g$ included |
| $\frac{4m_2u}{3}$ | A1 (3) | Must be positive and a single term. Allow fraction replaced by decimal to at least 2 SF |
### Part (b)
| Working/Answer | Mark | Guidance |
|---|---|---|
| CLM: $m_1u - m_2u = -m_1v + m_2\frac{1}{3}u$ **OR** $\frac{4m_2u}{3} = m_1(v - -u)$ | M1 A1 | M1 CLM with usual rules (allow consistent extra $g$'s), or impulse-momentum principle applied to $P$ using answer from (a). Must be in terms of $m_2$ and $u$ |
| $\frac{u(4m_2 - 3m_1)}{3m_1}$ oe | A1 (3) | Correct answer only. Any equivalent expression with $m_2$ terms collected. Allow fraction replaced by decimal to at least 2 SF. Allow consistent use of $-v$ instead of $v$ |
### Part (c)
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{u(4m_2 - 3m_1)}{3m_1} > 0$ | M1 | Correct inequality for their $v$, containing $u$. First statement must include $u$ **and** $> 0$ or $< 0$ as appropriate |
| $(4m_2 - 3m_1) > 0 \Rightarrow 4m_2 > 3m_1 \Rightarrow m_2 > \frac{3}{4}m_1$ | A1* (2) | Correct given answer correctly obtained. **N.B.** $\frac{3}{4}m_1 < m_2$ is A0 |
**N.B.** If they use $-v$ in (b), can score M1 for $-v < 0$ and possibly A1.
---\begin{enumerate}
\item Two particles, $P$ and $Q$, of mass $m _ { 1 }$ and $m _ { 2 }$ respectively, are moving on a smooth horizontal plane. The particles are moving towards each other in opposite directions along the same straight line when they collide directly. Immediately before the collision, both particles are moving with speed $u$.
\end{enumerate}
The direction of motion of each particle is reversed by the collision.\\
Immediately after the collision, the speed of $Q$ is $\frac { 1 } { 3 } u$.\\
(a) Find, in terms of $m _ { 2 }$ and $u$, the magnitude of the impulse exerted by $P$ on $Q$ in the collision.\\
(b) Find, in terms of $m _ { 1 } , m _ { 2 }$ and $u$, the speed of $P$ immediately after the collision.\\
(c) Hence show that $m _ { 2 } > \frac { 3 } { 4 } m _ { 1 }$
\hfill \mbox{\textit{Edexcel M1 2020 Q1 [8]}}