Mark Schemes for "Solomon" M2 practice papers A to F

Transcription

1 Mark Schemes for "Solomon" M practice papers A to F M Paper A Marking Guide. cons. of mom: m(5) m() m + m M + 5 ( ) 4 sole simul. giing m s - so speed is m s -, m s - (6). (a) dr dt (t )i + t j when t 0, i m s - at t, i + j ( + ) 5 M KE lost ()( 5) 6J (8). (a) (i) uniform rod B (ii) particle B S A R 5g 75g B µr resole : R 5g 75g 0 R 90g M resole : µr S 0 S 0g mom. about B S.8sinθ 5g.4cosθ 75g.dcosθ 0 8S tanθ 60g 75gd M 40g d 5.6 AP m (0) 75g 4. (a) a (t 5) a k(t 5) M a d t k(t 5t) + c when t 0, 0 so c 0 when t, so k(7 5) k 4 a 4 (t 5) s d t 4 ( 4 t 4 5 t ) + c when t 0, s 0 so c 0 s 4 ( 4 t 4 5 t ) M s 4 t ( 4 t 0 4 ) 6 t (t 0) () M A MARKS page

2 5. (a), portion mass x y mx my ABFG 56ρ ρ 4096ρ CDEF 8ρ ρ 5ρ total 84ρ x y 07ρ 4608ρ ρ mass per unit area x, y coords. taken horiz. / ert. from G M A 07 ρ x 8 so must lie on BF y 84ρ 4608ρ 84 ρ dist. from AB 0 cm (c) A θ 0 M 8 8 tanθ 0 θ.8 (dp) () P 6. (a) R ma a M a.5 m s - P at max. speed, a 0, R so 50 m s - KE J 500 kj (c) P R mgsinα (9.8)sinα 0 M.5 sinα 9. 8 α 8.8 (dp) () 7. (a) particle moing freely under graity B ert. disp. 0 t(u sinα gt) 0 M t 0 at O, we require 49sin0 4.9t 0 t 5 horiz. disp. utcosα 49(5)cos0.7 i.e. 70 beyond hole 4. m (sf) (c) when horiz. disp. 70, utcosα 70 t M horiz. el. ucosα 4.44 ert. el. usinα gt 4.76 A mag. of el [(4.44) + ( 4.76) ] 44.9 m s - (sf) req d angle tan below horizontal (sf) (5) Total (75) M A MARKS page

3 Performance Record M Paper A Question no Total Topic(s) collisions i, j calculus statics ladder prob. ariable accel. centre of mass power, KE projectiles Marks Student M A MARKS page 4

4 M Paper B Marking Guide. (a) work done force dist J work done change in KE m( u ) M 0.05( 00 ) M m s - (7). (a) P P at max. speed, a 0, R P W H P 7000 R ma 000 m m m 50 kg (7). (a) A 5 9 B C (i) c.o.m. dist. from B to C 9 cm from AB (ii) AB (5 9 ) cm c.o.m. dist. from B to A 4 cm from BC lamina will not topple if ertical through c.o.m. passes between B and C B max. θ when it passes through B B 4 θ tanθ 4 θ 6.9 (dp) (0) 4. (a) d a dt i tj and when t, a i 4j mag. of a [() + ( 4) ] 5 m s - s d t t i t j + Ai + Bj when t 0, s 6i + j so A 6, B s ( t + 6)i + ( t )j disp. when t 6 is 60i 60j 60(i j) k 60 (0) M M B MARKS page

5 5. (a) Y A X 8g mom. about A 8g(acos0 ) F(asin0 ) 0 F 4g ο tan 0 B F 08 N (sf) resole : F + X 0 X 08 resole : Y 8g 0 Y 78.4 N mag. of reaction at hinge [( 08) + (78.4) ] N (sf) req d angle tan - 54 (nearest degree) to the ertical (0) 6. (a) s y (utsinα gt ) t(usinα gt) u s y 0 when t 0 (at A) and when t g sinα (at B) u s x utcosα u( g sinα) cosα (at B) M u u g (sinα cosα) g sinα u sinα 80 g 45 sinα M sinα 0.87 giing α.4, 78.6 (dp) (c).4 as larger horiz. component of el. B 45 (d) t sin (.4 ).8 seconds (dp) (5) g 7. (a) cons. of mom: 4m(u) + 0 4m + 5m M 4u u 0 u sole sim. eqns. to get 6 u, u 6 4 u 4 u speed of B after collision with wall ue ue u 6 ue cons. of mom: 4m( 6 u) 5m( ue) 4mw + 0 u 0 ue 4w w u 0ue 0 w u+ ue 6 w u + ue giing w u + 4ue eliminating w gies u + 4ue + u 0ue 0 M u 6ue e (6) Total (75) M B MARKS page

6 Performance Record M Paper B Question no Total Topic(s) work - energy power centre of mass, toppling i, j calculus statics projectiles collisions Marks Student M B MARKS page 4

7 M Paper C Marking Guide dr. (a) dt 6ti 8tj a d dt 6i 8j not dependent on t so constant F ma a i 6j mag. of F [() + ( 6) ] 0 N (7). (a) X-sect. area of pipe πr π(0.05) mass of water per second π 000 5π energy gained m + mgh 5 π(6) + (50π 9.8 ) M 690 J 6.9 kj (sf) (8). (a) when t 0, 4 m s - particle at rest when t 9t i.e. (t )(t 4) 0 t, 4 (c) s d t t 9 t + 4t + c when t 0, s 9 so c 9 s t 9 t + 4t + 9 disp. when t 6 is (6) 9 (6) + 4(6) + 9 M m (9) 4. S S R mg R resole : S + R mg 0 M resole : R S 0 M sole simul. giing S R R 6 7 mg mom. about top of ladder R.acosθ R.asinθ mg.acosθ 0 tanθ R mg 5 R 4 M (9) M C MARKS page

8 5. (a) ert. disp. 0 8u y g(8) 0 u y g(8) 4g horiz. disp. 4 8u x 4 so u x initial speed [(4g) + ] 9. m s - (sf) (c) max. ht. when ert. el 0 0 (4g) gs max. ht. 8g 78.4 m (d) e.g. small X-section, reasonable to treat as particle and ignore air res. but, significant loss of mass during flight model not ery suitable B () 6. (a) cons. of mom: mu + 0 m + m M + u u u sole simul. giing u and u cons. of mom: mu + 0 mw + mw M w + w u w w u e w w eu sole simul. giing w u( e) A and B collide again so speed of B < speed of A M u( e) < u so e > e > (4) 7. (a) from triangle properties, area of BCD area of ABD B area of BCD ( d d) d portion mass y my ABD d ρ d d ρ BCD ABCD d ρ 9 d 9 d ρ d ρ y 8 d ρ 9 ρ mass per unit area y coords. taken ert. from BD M A y 8 9 d ρ d ρ 4d 9 4 d (c) A D θ B θ tan d tan - 4 d 9 req d angle 60 θ.4 (dp) (5) Total (75) M C MARKS page

9 Performance Record M Paper C Question no Total Topic(s) i, j calculus KE + PE ariable accel. statics ladder prob. projectiles collisions centre of mass Marks Student M C MARKS page 4

10 M Paper D Marking Guide dr. (a) dt (t )i + (t k)j M at rest when coeffs of i and j are both zero M t 0 t k 0 M both satisfied when k (6). cons. of mom: mu 5mu m() + 5m(4) u 5u 4 4 ( ) + u u u + u 4 sole simul. giing u m s -, u m s - (6). (a) R R k, where k is a constant M P R k 0 k 6 R 6 P R mgsinθ (9.8) quad. form. giing 9.7 m s - (sf) (clearly 6.0 not suitable) (0) 4. Y T A X B g 6g (a) mom. about A ga + 6g(a) Tacos60 0 4ga Ta T 8g resole : Y + Tcos60 8g 0 Y 6g resole : X Tsin60 0 X 4 g mag. of force at hinge [(4 g) + ( 6g) ] 45 N (sf) req d angle tan - 6g.9 (sf) below horizontal (away from wall) () 4 g 5. (a) a d t t 0t + c when t 0, so c t 0t + 0 when (t )(t ) 0 t, s d t t 5t + t + k when t 0, s 0 so k 0 s t 5t + t disp. when t is ( ) 5( ) + ( ) 7 disp. when t is () 5() + () 6 dist. traelled m () M D MARKS page

11 6. (a) min. α when ball passes through (, 0.6) 4tcosα 6 t 7 cosα 0.6 4tsinα 4.9t M sub. in t giing ( )sinα 4.9( ) 7 cosα 7 cosα 0.6 tanα.6sec α M use sec α + tan α giing 6tan α 0tanα use of quad. form. giing tanα 0.7 (and.06) min. α 5 (nearest degree) utcosα 4t( 0 5 ) t 7 ert. disp., utsinα gt 4( 7 )( 4 5 ) 4.9( 7 ) M i.e aboe M m aboe crossbar (4) 7. (a) (i), (ii) portion mass x y mx my rectangle ρ 4 8ρ 64ρ semicircle πρ π πρ (8π + )ρ total ( + π)ρ x y (8 + π)ρ (8π + )ρ ρ mass per unit area x, y coords. taken horiz. / ert. from O M4 A (8+ π ) ρ x 4. cm from OD (sf) y (+ π ) ρ 08 (8π + ) ρ (+ π ) ρ.47 cm from OA (sf) from m pt. of BC ertically M from m pt. of BC horizontally M.5 θ.67.5 tanθ θ 4.5 (sf). 67 (4) Total (75) M D MARKS page

12 Performance Record M Paper D Question no Total Topic(s) i, j calculus collisions power statics ariable accel. projectiles centre of mass Marks Student M D MARKS page 4

13 M Paper E Marking Guide. I mom. i 9j 0.6[(5i + j) u] 0i 5j 5i + j u M u 5i +8j (4). (a) when t 0, x m d x dt 0 e t at rest when 0 0 e t 0 e t 0 t ln 0. (dp) (6). (a) µs S R 0g T B resole : R + µs 0g 0 R 0g µs M resole : T S 0 S T M eliminating S gies R 0g T mom. about top of ladder T(4sinθ) + 0g(cosθ) R(6cosθ) 0 4Ttanθ + 60g 6R 0 M 0T + 60g 0g + T 0 T 60g and T 5g (c) attach rope lower down ladder/wall B gies larger moment about top of ladder with same tension B () 4. (a) (i), (ii) portion mass x y mx my AB aρ 0 a 0 a ρ BC aρ a 0 9 a ρ 0 CD aρ a total 6aρ x y a a ρ 5 a ρ a ρ 5 a ρ ρ mass per unit area x, y coords. taken horiz./ ert. from B M A 5 a ρ x 6aρ 4 from AB 5 a ρ y ρ from BC 6a a 5a 9a A θ 9a tanθ 5 a 4 9 a 5a θ 8 (nearest degree) M () M E MARKS page

14 P 5. (a) R mgsinα 0 P (9.8) M P 0( ) W 480 kw P R ma a 0 a 0.49 m s - (c) at max. speed, a 0 P R 0 M so 09 m s - (sf) (d) model not suitable lorry unable to attain 09 m s - ( 45 mph) B () 6. (a) cons. of mom: M(U) + 0 M(V) + 5M(4) M U V V 4 V 4 U U 0 4 sole simul. giing U 8 s y gt 9.6, t 4 t M (c) x 4, y 0 gt 9.6 req d angle tan (sf) below horizontal () 4 7. (a) R µr α mg m mass of P d AB resole perp. to plane: R mgcosα 0 R mg( 5 ) frictional force µr 5 mg work done against friction loss in KE gain in PE M mgd m(5.6) mgdsinα 5.68m 4 5 mgd M A 5 40 gd 5 (5.6) d.4 m work done against friction loss in KE (as PE returns to initial alue) mg.8 5 m(5.6 ) M.9g 5.6 M m s - (sf) (6) Total (75) M E MARKS page

15 Performance Record M Paper E Question no Total Topic(s) i, j impulse ariable accel. statics centre of mass power collisions, projectiles work - energy Marks Student M E MARKS page 4

16 M Paper F Marking Guide. I mom. 0.5[(i + 7j) (5i 8j)] 0.5(8i + 5j) mag. of I 0.5 (8 + 5 ) 8.5 N s (5). (a) change in KE 000(0 0 ) J change in PE 000(9.8)(00sinθ) J M change in ME increase of 0000 J air resistance B friction B (8). (a) s t(t t + 0) t(t 5)(t 4) particle at O when s 0 at t 0, 5, 4 seconds d s at rest when 0, dt 6t 6t + 0 t t + 0 0, (t )(t 0) 0 M t, seconds (8) 4. (a) S µs B R 6g A mom. about B: 6gcos0 R.cos0 0 R g mom. about A: 6gcos0 S. 0 S g resole : µssin60 Ssin0 0 µ (9) 5. (a) at max. ht., y 0 0 ( sinα) gs 7 (. ) 8 s y 8.9 M g starts.6 m aboe P so max. ht. aboe ground 0.5 m (sf) s y.4 utsinα gt.4 77 t 4.9t.4 4 4t 55t 4 0 (4t + )(t 4) 0 M t 4 in this case ball in flight for 4 seconds (c) s x utcosα max. dist. fielder can run is m max. initial dist. between fielder and ball m (sf) () M F MARKS page

17 6. (a) (c) a,since masses on AD are equal to mass at B portion mass y my lamina 8m a 8ma particle at A m 0 0 particle at B 6m 0 0 particle at D 4m a 8ma total 0m y 6ma y coords. taken ert. from AB M 6 ma 4 y a 0m 5 portion mass x mx lamina 8m 4ma particle at A m 0 0 particles at B (6 + k)m a (6 + k)ma particle at D 4m 0 0 total (0 + k)m x (0 + k) ma a x coords. taken horiz. from AD (0+ k ) ma (0+ k) a x (0+ k ) m (0+ k) (d) new y 6ma (0+ k) m 6a (0+ k) M 45 (0+ k) a (0+ k) tan 45 6a (0+ k) 6a (0+ k ) a 6 giing k 6 M (6) 0+k u 7. (a) cons. of mom: 7u + 4u 7( ) + 4 M 8 7u + 8u u u u e eu eu + u eliminate giing 7u + 8u 8eu 8eu + 4u 8u + 8eu 8eu u 8u (e + ) u (8e ) sub. in for u and u : 4(e + ) 4(8e ) M 4e + 4 e 4 giing e 4 (c) speeds of A, B after impact are and resp. 7 m s -, ( 8 7 ) m s - original KE J final KE J % KE lost % (sf) (7) 704 Total (75) M F MARKS page

18 Performance Record M Paper F Question no Total Topic(s) i, j impulse energy ariable accel. statics projectiles centre of mass collisions, energy Marks Student M F MARKS page 4