A-LevelMathematicsMechanics9709/51 · Oct/Nov 2017 · Q58 Marks

One end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N is attached to a fixed point O. The other end of the string is attached to a particle P of mass 0.3 kg. P is projected vertically upwards with speed 4 m s⁻¹ from a position 1.2 m vertically below O.

(i)Calculate the speed of the particle at the position where it is moving with zero acceleration.[5]
(ii)Show that the particle moves 1.2 m while moving upwards with constant deceleration.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[M1] 0.3g = 24e Use T = λx/L
[A1] e = 0.1
[B1] EE =24 × (1.2-0.8)²/(2 × 0.8) or 24 × 0.1²/(2 × 0.8) Use EE = λ x² /(2L).
[M1] 0.3 v² /2 = 0.3 × 4² /2 + 24 × (1.2 – 0.8)²/(2 × 0.8) -24 × 0.1²/(2 × 0.8) – 0.3g(1.2 – 0.8) Sets up a 5 term energy equation involving EE, KE and PE.
[A1] v = 5 ms⁻¹
[M1] 0.5 × 5² /2 + 24 × 0.1²/(2 × 0.8) = 0.3(x + 0.9) ×10 Sets up a 3 term energy equation where x is the distance above 0 when v = 0.
[A1] x = 0.4
[A1] Distance moved = 0.8 + 0.4 = 1.2 m AG

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