M1 June 2016 Q3
3. A particle \(P\) of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, \(P\) is moving with speed 4 m s\(^{-1}\) in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between \(P\) and the ground is \(\dfrac{1}{8}\)
Find the magnitude of the impulse exerted on \(P\) by the wall. (7)
| Scheme | Marks |
|---|---|
| \(F = \tfrac{1}{8} \times 0.4g\) | M1 |
| \(-\tfrac{1}{8} \times 0.4g = 0.4a\) | M1 A1 |
| \(0 = u^2 + 2\left(-\tfrac{1}{8}g\right) \times 5\) | M1 A1 |
| \(I = 0.4 \times (3.5 - -4) = 3\) Ns | M1 A1 |
| (7 marks) |
Notes
First M1 for \(1/8 \times 0.4g\) (Allow if \(g\) omitted)
Second M1 for resolving horizontally with their \(F\) (could just be \(F\))
First A1 for a correct equation in \(a\) only
Third M1 for use of \(v^2 = u^2 + 2as\) with \(v = 0\), \(s = 5\) and a calculated value of \(a\). (M0 if \(u = 4\) or if \(u = 0\))
Second A1 for a correct equation in \(u\) only (\(u\) may be in terms of \(I\))
Fourth M1 (M0 if \(g\) included or if \(u = 0\) or \(u = 4\)) for \(\pm 0.4(u - \pm 4)\) where \(u\) is their calculated value.
Third A1 for 3, 3.0 or 3.00 (Ns)
Alternative work –energy method:
| \(F = (\tfrac{1}{8} \times 0.4g)\) | M1 |
| \(\tfrac{1}{2}0.4u^2 = (\tfrac{1}{8} \times 0.4g) \times 5\) | M2 A2 (M2 if \(F\) not substituted) |
| \(I = 0.4 \times (3.5 - -4)\) | M1 |
| \(= 3\) (Ns) | A1 |