Moderate -0.3 This is a straightforward application of conservation of momentum in 2D with coalescing particles. Students apply momentum conservation separately to each component, solve for m from the j-component equation, then substitute to find k. While it involves vectors, it requires only routine algebraic manipulation with no conceptual challenges or novel problem-solving.
4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg
Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant.
Find the value of \(k\).
Forms conservation of momentum equation: \(m\begin{bmatrix}4\\-2\end{bmatrix} + 2\begin{bmatrix}-3\\5\end{bmatrix} = (m+2)\begin{bmatrix}k\\0\end{bmatrix}\)
M1
Either \(4m + 2(-3) = k(m+2)\) or \(-2m + 2(5) = 0\); can be embedded in single 2D equation
Two fully correct equations: \(-2m + 10 = 0\) and \(20 - 6 = 7k\)
A1
Obtains \(m = 5\)
A1
Deduces \(k = 2\) using their \(m\) and a correct momentum equation
R1F
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Forms conservation of momentum equation: $m\begin{bmatrix}4\\-2\end{bmatrix} + 2\begin{bmatrix}-3\\5\end{bmatrix} = (m+2)\begin{bmatrix}k\\0\end{bmatrix}$ | M1 | Either $4m + 2(-3) = k(m+2)$ or $-2m + 2(5) = 0$; can be embedded in single 2D equation |
| Two fully correct equations: $-2m + 10 = 0$ and $20 - 6 = 7k$ | A1 | |
| Obtains $m = 5$ | A1 | |
| Deduces $k = 2$ using their $m$ and a correct momentum equation | R1F | |
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4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg
Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$
As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant.
Find the value of $k$.\\
\hfill \mbox{\textit{AQA Further AS Paper 2 Mechanics 2020 Q4 [4]}}