A-LevelMathematicsMechanics9709/42 · May/June 2023 · Q611 Marks

A particle P starts at rest and moves in a straight line from a point O. At time ts after leaving O, the velocity of P, vms¯¹, is given by v = bt + ct³, where b and c are constants. P has velocity 8ms⁻¹ when t = 4 and has velocity 13.5ms¯¹ when t = 9.

(a)Show that b = 3 and c = -0.5.[1]
(b)Find the acceleration of P when t = 1.[2]
(c)Find the positive value of t when P is at instantaneous rest and find the distance of P from O at this instant.[5]
(d)Find the speed of P at the instant it returns to O.[3]

Mark scheme

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

6(a)
4b + 4²c = 8[→ 4b + 8c =8] and 9b +9²c=13.5[→9b+27c=13.5] State b=3 and c = -0.5
[B1] Must have 2 correct equations, which do not have to be simplified. Allow to just state the values of b and c. Allow substitution of b = 3 and c = -0.5 in both equations to verify. No further calculation required. B0 if any incorrect work seen.
OR 3×4 + (-0.5)×4² = 8 AND 3×9+(-0.5)×9² =13.5
6(b)
a = dv/dt = (3 - 0.5 × 3/2 × t²) OR b + c × 3/2 × t²
[M1] Attempt to differentiate, decrease power by 1 and a change in coefficient in at least one term (which must be the same term); allow unsimplified; a = v/t is M0.
acceleration = 2.25 ms⁻²
[A1] OE, e.g. 9/4 or 2¼. SC B1 for 2.25 if no differentiation seen.
6(c)
Equate v to 0 and attempt to solve for t
[M1] Must get to t = ... and must be positive.
t = 36 ONLY
[A1] WWW. Allow if t = 0 seen and not rejected.
Attempt to integrate
[M1] Increase power by 1 and a change in coefficient in at least one term (which must be the same term). s = vt is M0.
s = (3/2)t² - (0.5/4)t⁴ OR s = (b/2)t² + (c/4)t⁴
[A1] Allow unsimplified (including indices).
Sub t = 36 (or use limits 36 and 0) to get distance = 388.8 m ONLY
[A1] Allow 389 m. If no integration seen for the last 3 marks, allow SC B1 for 388.8 m. Max M1A1B1 for 3/5 marks.
6(d)
Equate their s (that has come from an integration attempt) to 0 and attempt to solve for t. Must get to t = ...
[M1]
t = 56.25
[A1] WWW. OE, e.g. 225/4. Allow 2sf or better. Allow any correct unsimplified expression equivalent to 56.25 e.g. 15²/2² or 7.5².
Speed = 42.2 ms⁻¹ ONLY
[A1] AWRT 42.2. Speed = -42.2 is A0. Allow A1 if negative sign dropped without justification.