A2 June 2024 Q1
1. [In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal perpendicular unit vectors.]
A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg.
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \((3\mathbf{i} - \mathbf{j})\ \text{m s}^{-1}\) and the velocity of \(B\) is \((-6\mathbf{i} + 2\mathbf{j})\ \text{m s}^{-1}\)
Immediately after the collision the velocity of \(A\) is \(\left(-2\mathbf{i} + \dfrac{2}{3}\mathbf{j}\right)\ \text{m s}^{-1}\)
| Scheme | Marks | AO |
|---|---|---|
| Form an expression for total KE | M1 | 3.1a |
| \(\dfrac{1}{2} \times 3 \times (3^2 + (-1)^2) + \dfrac{1}{2} \times 2 \times ((-6)^2 + 2^2)\) | A1 | 1.1b |
| \(= 55\) (J) | A1 | 1.1b |
| (3) |
Notes
Ignore units in final answers
M1: Form an expression for total KE, correct number of terms and dimensionally correct. Must add 2 KE terms of the correct structure. Allow missing minus signs on components when squaring.
M0 for use of \((3\mathbf{i} - \mathbf{j})^2\) unless recovered.
M0 if KE terms are never added together.
M0 for using \(\tfrac{1}{2}mv\) unless the correct KE formula is also stated. (Note that \(\tfrac{1}{2}mv\) is not dimensionally correct)
A1: Correct unsimplified expression for the total KE. If the 2 KE terms are incorrect when the total is formed, this is A0.
A1: Cao
| Scheme | Marks | AO |
|---|---|---|
| Use of change in momentum | M1 | 3.1a |
| \(3\left(\left(-2\mathbf{i} + \dfrac{2}{3}\mathbf{j}\right) - (3\mathbf{i} - \mathbf{j})\right)\) | A1 | 1.1b |
| \(= (-15\mathbf{i} + 5\mathbf{j})\) (N s) | A1 | 1.1b |
| (3) |
Notes
Ignore units in final answers
N.B. Only penalise use of column vectors in final answers once across (b) and (c). Penalise the first occurrence. Allow column vectors throughout working in (b) and (c).
M1: Use of change in momentum, dimensionally correct (mass \(\times\) velocity). Must subtract momenta but condone in wrong order. Must use mass of 3kg with velocities of \(A\) or mass of 2kg with velocities of \(B\).
M0 if speeds are used.
A1: Correct unsimplified expression for impulse
A1: Correct answer in i and j form. Penalise first occurrence if answer is given as a column vector. ISW if they continue and find the magnitude.
| Scheme | Marks | AO |
|---|---|---|
| Either: Use impulse on \(B\) to form equation | M1 | 3.1a |
| \((15\mathbf{i} - 5\mathbf{j}) = 2\big(\mathbf{v}_B - (-6\mathbf{i} + 2\mathbf{j})\big)\) | A1 | 1.1b |
| \(\mathbf{v}_B = (1.5\mathbf{i} - 0.5\mathbf{j})\ (\text{m s}^{-1})\) | A1 | 1.1b |
| Or: Use CLM to form equation | M1 | 3.1a |
| \(3(3\mathbf{i} - \mathbf{j}) + 2(-6\mathbf{i} + 2\mathbf{j}) = 3\left(-2\mathbf{i} + \dfrac{2}{3}\mathbf{j}\right) + 2\mathbf{v}_B\) | A1 | 1.1b |
| \(\mathbf{v}_B = (1.5\mathbf{i} - 0.5\mathbf{j})\ (\text{m s}^{-1})\) | A1 | 1.1b |
| (3) | ||
| (9 marks) |
Notes
Ignore units in final answers
N.B. Only penalise use of column vectors in final answers once across (b) and (c). Penalise the first occurrence. Allow column vectors throughout working in (b) and (c).
M1: Either: Use of impulse-momentum equation with their Impulse. Must subtract momenta (mass \(\times\) velocity) but condone in wrong order. Condone sign error on LHS.
A1: Correct unsimplified equation
A1: Correct answer in i and j form. Penalise first occurrence if answer is given as a column vector.
M1: Or: Use of CLM equation with correct number of terms, condone sign errors. Masses and velocities must be paired correctly. M0 if speeds are used.
A1: Correct unsimplified equation
A1: Correct answer in i and j form. Penalise first occurrence if answer is given as a column vector.