Two particles A and B, of masses 0.4kg and 0.2kg respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at 30° to the horizontal, and A is higher up the plane than B. When the particles collide, the speeds of A and B are 3ms¯¹ and 2ms¯¹ respectively. In the collision between the particles, the speed of A is reduced to 2.5 ms¯¹.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [M1] Use of conservation of momentum with 4 terms. Allow sign errors.
2. [A1] v = 3 ms⁻¹ (Allow M1A0 if g included with the masses.)
3. [M1] For either A ±0.4g sin 30° = 0.4a or for B ±0.2g sin 30° = 0.2a or ±mg sin 30° = ma (Allow sin/cos mix.)
4. [A1] a = ±5 or ±g sin 30° (Allow g sin 30° without working for M1A1)
5. [*M1] For B when hits barrier v² = 3² + 2*5*1.6 [=> v=5] OR v=u+at => v=3+5*0.4 [=> v = 5] (Using their a != ±g and their v from part (a)) (OR: use of s = (u+v)/2 * t => 1.6 = (3+v)/2 * 0.4 [=>v=5]) (OR 1/2*0.2x² - 1/2*0.2*3² = 0.2*1.6*gsin 30)
6. [A1] Speed after hitting barrier = 0.1 * 5 = 0.5 (AG)
7. [DM1] VA = 2.5 + 5 * 0.44 [= 4.7] vB = -0.5 + 5 * 0.04 [= -0.3] or vB = 0.5 + (-5) * 0.04 [= 0.3] (Use of v=u+at for BOTH with correct t-values, initial speeds ±2.5,±0.5 and ± their acceleration (same for both) and use of conservation of momentum with correct number of terms. Allow sign errors.)
8. [A1] 0.4 * 4.7 + 0.2 * (-0.3) = 0.6 Vcomb
9. [A1] Vcomb = 3.03 ms⁻¹ (Allow v = 91/30 = 3 1/30. Allow DM1A0 if g included with the masses.)