Particle P of mass m kg and particle Q of mass 2 kg are free to move on a smooth horizontal plane. P and Q are moving directly towards each other with speeds u ms⁻¹ and 3u ms⁻¹ respectively. P and Q collide and the direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of P is (1/3)u ms⁻¹.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [1] Attempt at conservation of momentum with correct number of terms. Allow sign errors ONLY so no misreads in this part. However, if using g with either (or both) masses, then allow this mark but do not allow subsequent A marks.
2. [1] A correct (un-simplified) expression for the velocity of Q after the collision e.g. (2m-9)u/6, (mu/3)-3u/2, (mu-6u+1/2 mu)/2 etc.
3. [1] AG a correct expression for v put > 0 followed by m>4.5. Note that (2/3)mu-3u=0⇒m=4.5 ∴ m > 4.5 is A0. Using trial and improvement to show that v is negative with value(s) of m that are less than 4.5 is A0 unless they also show that v = 0 when m = 4.5. Similarly, if attempting to show that v is positive with value(s) of m that are greater than 4.5 they must also show that v = 0 when m = 4.5 for A1. This A1 mark is dependent on the previous A1 mark.
4. [1] [Q's speed is reduced by 75% by its impact with the wall ⇒ ] speed after impact is (1/4) * |(2/3)mu - 3u|. Follow through their v of the form k₁mu + k₂u (if simplified) for k1,k2 ≠ 0 from 6(a) OE e.g. (2m-9)u/12 or possibly implied by later working e.g. -u/3 < (9-2m)u/12. Allow + e.g. (2/3 mu - 3u)/4. Look out for this expression embedded in later (possibly incorrect) working.
5. [1] No further collisions therefore -(1/4)u((2/3)m - 3) >= -(1/3)u. For -(1/3)u > v with v of the form k₁mu + k₂u for k1,k2 ≠ 0 from 6(a) - allow > or = for this mark Note that (4/3)u = (2/3)mu-3u (which leads to m = 6.5) scores B1 M1 ONLY.
6. [1] Solve inequality to obtain m ≤ 6.5. Dependent on both previous marks - M0 if an equation for m found rather than an inequality for a range of values of m.
7. [1] (4.5<)m ≤ 6.5 ⇒ m=6 CWO – condone m = 6 from m < 6.5.