M1 June 2017 Q2
2. Two particles, \(P\) and \(Q\), have masses \(2m\) and \(3m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane when they collide directly. Immediately before they collide the speed of \(P\) is \(4u\) and the speed of \(Q\) is \(3u\). As a result of the collision, \(Q\) has its direction of motion reversed and is moving with speed \(u\).
| Scheme | Marks |
|---|---|
| \(8mu - 9mu = -2mV + 3mu\) | M1 A1 |
| \(V = 2u\) | A1 |
| (3) |
Notes
M1 for CLM with correct no. of terms, all dimensionally correct, to give an equation in \(m\), \(u\) and their \(V\) only. Condone consistent \(g\)’s or cancelled \(m\)’s.
First A1 for a correct equation (they may have \(+\,2mV\))
Second A1 for \(2u\) (must be positive since speed is required)
| Scheme | Marks |
|---|---|
| (Has been) reversed | B1 |
| (1) |
Notes
B1 for ‘(has been) reversed’. Only available if a correct velocity has been correctly obtained in part (a).
B0 for ‘changed’, ‘direction has changed’, ‘yes’
| Scheme | Marks |
|---|---|
| For \(Q\): \(\ I = 3m(u - -3u)\) | M1 A1 |
| \(= 12mu\) | A1 |
| OR: | OR |
| For \(P\): \(\ I = 2m(2u - -4u)\) | M1 A1 |
| \(= 12mu\) | A1 |
| (3) | |
| (7 marks) |
Notes
M1 for using Impulse = change in momentum of \(Q\) (must have \(3m\) in both terms) (M0 if clearly adding momenta or if \(g\) is included) but condone sign errors.
First A1 for \(3m(u - -3u)\) or \(-3m(u - -3u)\)
Second A1 for \(12mu\) (must be positive since magnitude required)
OR
M1 for using Impulse = change in momentum of \(P\) (must have \(2m\) in both terms) (M0 if clearly adding momenta) but condone sign errors.
First A1 for \(2m(2u - -4u)\) or \(-2m(2u - -4u)\)
Second A1 for \(12mu\) (must be positive since magnitude required)
N.B. Allow use of \(I = 3m(u - v)\) or \(I = 2m(u - v)\) since only magnitude required