A-LevelMathematicsMechanics9709/42 · Feb/Mar 2023 · Q713 Marks

The diagram shows a smooth track which lies in a vertical plane. The section AB is a quarter circle of radius 1.8 m with centre O. The section BC is a horizontal straight line of length 7.0m and OB is perpendicular to BC. The section CFE is a straight line inclined at an angle of θ° above the horizontal. A particle P of mass 0.5kg is released from rest at A. Particle P collides with a particle Q of mass 0.1 kg which is at rest at B. Immediately after the collision, the speed of P is 4ms⁻¹ in the direction BC. You should assume that P is moving horizontally when it collides with Q. [Figure X.X]

(a)Show that the speed of Q immediately after the collision is 10ms⁻¹.[4]
(b)When Q reaches C, it collides with a particle R of mass 0.4kg which is at rest at C. The two particles coalesce. The combined particle comes instantaneously to rest at F. You should assume that there is no instantaneous change in speed as the combined particle leaves C, nor when it passes through C again as it returns down the slope. Given that the distance CF is 0.4 m, find the value of θ.[4]
(c)Find the distance from B at which P collides with the combined particle.[5]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.

[M1] Attempt to use conservation of energy 2 terms, dimensionally correct.
[A1] v=6 Do not allow from use of constant acceleration.
[M1] Attempt at conservation of momentum 3 terms; allow sign errors; allow their v=6 or just v ; allow if using mgv (consistently in all terms).
[A1] Speed of Q (=w)=10 ms⁻¹ AG
[*M1] Attempt at conservation of momentum 3 terms, allow sign errors, allow if using mgv.
[*DM1] Attempt to use conservation of energy Dependent on previous M mark. 4 terms, dimensionally correct.
[DM1] Use trigonometry to get an equation in θ and solve for θ Dependent on previous 2 M marks. Using their h and 0.4 . Allow sin/cos mix.
[A1] θ=30
[B1] Where b is distance from B
[M1] Attempt to form an equation from use of total time and solve for b (or c) Where b is distance from B Must have considered all parts of motion to find times from relevant equations.
[A1] Distance from B = 6 ⅔ m

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.