M4 June 2013 Q7
7. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane]
A small smooth ball of mass \(m\) kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector \(2\mathbf{i} + \mathbf{j}\). The velocity of the ball immediately before the impact is \(b\mathbf{i}\) m s\(^{-1}\), where \(b\) is positive. The velocity of the ball immediately after the impact is \(a(\mathbf{i} + \mathbf{j})\) m s\(^{-1}\), where \(a\) is positive.
Find
| Scheme | Marks |
|---|---|
| State that impulse acts perpendicular to the wall and demonstrate that \((2\mathbf{i} + \mathbf{j}).(-\mathbf{i} + 2\mathbf{j}) = 0\) | B1 |
Notes
B1 Requires scalar product or gradient diagram.
| Scheme | Marks |
|---|---|
| Impulse momentum equation: \(m(\mathbf{v} - \mathbf{u}) = m[(a - b)\mathbf{i} + a\mathbf{j}] = \lambda(-\mathbf{i} + 2\mathbf{j})\) | M1 A2 |
| \(\Rightarrow a = -2(a - b),\ \ 3a = 2b\) | A1 |
| OR | |
| Taking scalar products of velocities with \((2\mathbf{i} + \mathbf{j})\) | M1 |
| \(\begin{pmatrix}b\\0\end{pmatrix}\bullet\begin{pmatrix}2\\1\end{pmatrix} = 2b\) and \(\begin{pmatrix}a\\a\end{pmatrix}\bullet\begin{pmatrix}2\\1\end{pmatrix} = 3a\) | A1A1 |
| No change parallel to the wall so \(2b = 3a\). | A1 |
| Scalar products with \((-\mathbf{i} + 2\mathbf{j})\): \(\begin{pmatrix}b\\0\end{pmatrix}\bullet\begin{pmatrix}-1\\2\end{pmatrix} = -b\) and \(\begin{pmatrix}a\\a\end{pmatrix}\bullet\begin{pmatrix}-1\\2\end{pmatrix} = a\) | B1 |
| Impact equation: \(a = eb\) | M1A1 |
| \(e = \dfrac{2}{3}\) | A1 |
Notes
M1 Requires all terms present and of the correct structure
A2 -1 each error
7(b) alt
| \(b\cos\theta = a\sqrt{2}\cos(45 - \theta)\) | M1 A2 |
| \(b\cos\theta = a\cos\theta + a\sin\theta,\ \ 2b - 2a = a\) | |
| \(2b = 3a\) | A1 |
| Use of \(\tan\theta = \dfrac{1}{2}\) | B1 |
| \(a\sqrt{2}\sin(45 - \theta) = eb\sin\theta\) | M1 |
| \(a\cos\theta = (a + eb)\sin\theta,\ \ 2a = a + eb\) | A1 |
| \(e = \dfrac{2}{3}\) | A1 |
M1 A2 Parallel to the wall. Condone trig confusion? -1 each error. Both angles in same variable?
B1 When seen in (b). Implied by 26.6 or 18.4
M1 Perpendicular to the wall. Condone consistent trig confusion?
A1 \(e = \sqrt{\dfrac{10a^2}{b^2} - 4}\)
A1 0.67 or better
| Scheme | Marks |
|---|---|
| Fraction of KE lost \(= \dfrac{b^2 - 2a^2}{b^2}\) | M1A1 |
| \(= \dfrac{1 - 2 \times \dfrac{4}{9}}{1} = \dfrac{1}{9}\) | A1 |
| (12 marks) |