M2 June 2018 Q5
5. A particle \(A\) of mass \(3m\) is moving in a straight line with speed \(2u\) on a smooth horizontal floor. Particle \(A\) collides directly with another particle \(B\) of mass \(2m\) which is moving along the same straight line with speed \(u\) but in the opposite direction to \(A\). The coefficient of restitution between \(A\) and \(B\) is \(\dfrac{1}{3}\).
After the collision, \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\dfrac{1}{2}\). The first collision between \(A\) and \(B\) occurred at a distance \(x\) from the wall. The particles collide again at a distance \(y\) from the wall.
| Scheme | Marks |
|---|---|
| CLM: \(\ 3m \times 2u - 2m \times u = 3mv + 2mw\) \((4u = 3v + 2w)\) | M1A1 |
| Impact law: \(\ w - v = \dfrac{1}{3}(2u + u) = u\) | M1A1 |
| Solve for simultaneous equations for \(w\) or \(v\): \(3w - 3v = 3u,\ \ 2w + 3v = 4u\) \(5w = 7u,\ \ w = \dfrac{7}{5}u\) | DM1 A1 |
| \(v = \dfrac{2}{5}u\) | A1 |
| (7) |
Notes
DM1 A1 Dependent on both previous M marks. Must see working - Given Answer
A1 Or equivalent. Must be positive
| Scheme | Marks |
|---|---|
| Speed of B after collision with wall: \(\dfrac{1}{2} \times \dfrac{7}{5}u = \dfrac{7}{10}u\) | B1 |
| Total time for either particle | B1 |
| Equate the time travelled for each particle: | M1 |
| \(\dfrac{x}{\frac{7}{5}u} + \dfrac{y}{\frac{7}{10}u} = \dfrac{x - y}{\frac{2}{5}u}\) | A1 |
| \(\dfrac{5x}{7u} + \dfrac{10y}{7u} = \dfrac{5x}{2u} - \dfrac{5y}{2u},\ \ \ \ 10x + 20y = 35x - 35y\) | DM1 |
| \(55y = 25x,\ \ \ \ \ y = \dfrac{5}{11}x\) | A1 |
| (6) | |
| (13 marks) |
Notes
B1 Accept +/-
A1 Correct unsimplified
DM1 Dependent on previous M1
A1 Or equivalent. 0.45x or better
Alt 1
| Speed of B after collision with wall: \(\dfrac{1}{2} \times \dfrac{7}{5}u = \dfrac{7}{10}u\) | B1 |
| Time of travel for B: \(\ \dfrac{x}{\frac{7}{5}u} + \dfrac{y}{\frac{7}{10}u} = \dfrac{5x + 10y}{7u}\) | B1 |
| Distance moved by A: | M1 |
| \(= \dfrac{2}{5}u \times \left(\dfrac{5x + 10y}{7u}\right) = \dfrac{2x + 4y}{7}\) | A1 |
| \(\dfrac{2x + 4y}{7} + y = x,\ \ 2x + 4y + 7y = 7x\) | DM1 |
| \(y = \dfrac{5}{11}x\) | A1 |
B1 Accept +/-
M1 Correct method for distance
A1 Correct unsimplified
DM1 Dependent on previous M1. Form equation in \(x\) and \(y\)
A1 Or equivalent. 0.45x or better
Alt 2
| Speed of B after collision with wall: \(\dfrac{1}{2} \times \dfrac{7}{5}u = \left(\dfrac{7}{10}u\right)\) | B1 |
| \(x -\) distance moved by \(A = x - \dfrac{2}{5}u \times \dfrac{5x}{7u} = \dfrac{5}{7}x\) | B1 |
| Gap closing at \(\ \dfrac{7}{10}u + \dfrac{2}{5}u = \dfrac{11}{10}u\) | |
| Time to collision: \(\ \left(\dfrac{5}{7}x\right) \div \left(\dfrac{11}{10}u\right) = \dfrac{50x}{77u}\) | M1A1 |
| Distance moved by \(B\): \(\ y = \dfrac{7}{10}u \times \dfrac{50x}{77u} = \dfrac{5}{11}x\) | DM1 A1 |
B1 Accept +/-
B1 Distance apart when \(B\) hits the wall
M1A1 Use of \(\dfrac{9x}{7}\) for \(\dfrac{5x}{7}\) is M0
DM1 A1 Dependent on previous M1. Or equivalent. 0.45x or better
Alt 3
| Speed of B after collision with wall: \(\dfrac{1}{2} \times \dfrac{7}{5}u = \left(\dfrac{7}{10}u\right)\) | B1 |
| \(x -\) distance moved by \(A = x - \dfrac{2}{5}u \times \dfrac{5x}{7u} = \dfrac{5}{7}x\) | B1 |
| Ratio of speeds 4:7 | M1A1 |
| Distance moved by \(B\): \(\ y = \dfrac{7}{11} \times \dfrac{5x}{7} = \dfrac{5}{11}x\) | DM1 A1 |
B1 Accept +/-
B1 Distance apart when \(B\) hits the wall
DM1 A1 Dependent on previous M1. Use of \(\dfrac{9x}{7}\) for \(\dfrac{5x}{7}\) is M0. Or equivalent. 0.45x or better
Alt 4
| Speed of B after collision with wall: \(\dfrac{1}{2} \times \dfrac{7}{5}u\left(= \dfrac{7}{10}u\right)\) | B1 |
| \(x -\) distance moved by \(A = x - \dfrac{2}{5}u \times \dfrac{5x}{7u} = \dfrac{5}{7}x\) | B1 |
| Equate times for each particle to cover the residual distance. \(\dfrac{5}{2u}\left(\dfrac{5x}{7} - y\right) = \dfrac{10}{7u} \times y,\ \ \dfrac{1}{2}\left(\dfrac{5x}{7} - y\right) = \dfrac{11}{7}y\) | M1A1 |
| Distance moved by \(B\): \(\ y = \dfrac{5}{11}x\) | DM1 A1 |
B1 Accept +/-
B1 Distance apart when \(B\) hits the wall
M1A1 Use of \(\dfrac{9x}{7}\) for \(\dfrac{5x}{7}\) is M0
DM1 A1 Dependent on previous M1. Or equivalent. 0.45x or better