CAIE Further Paper 3 (Further Paper 3) 2020 Specimen

1 A ch ld's ty co ists

2 A lig elastic strig hsa tn al leg h \(a\) ad md B 6 elasticity 2 mg . Or ed 6 th strig is attach d œ fiæ \(\Phi \quad n A\). Tb ob rend te strig s attach d œ \(\mathbf { P }\) rticle 6 mass \(2 m\).

1. Fid in terms \(6 a\), th ex en in 6 th strig wh \(n\)th \(\boldsymbol { \rho }\) rticle \(\mathbf { h }\) g freely in eq lib im b low \(A\).
2. Th \(\mathbf { p }\) rticle is released rm rest at \(A\). Fid in terms \(\mathbf { 6 } \quad a\), th \(\dot { \mathbf { d } }\) stance

3 A \(\boldsymbol { \rho }\) rticle \(P 6\) mass \(m\) g falls frm rest \(\mathbf { d } \quad \mathrm { rg }\) avty. The re is a resistiv fo ce 6 mag td \(m k v ^ { 2 } \mathrm {~N}\), wh re \(v \mathrm {~ms} ^ { - 1 }\) is th se e \(\boldsymbol { C }\) after it \(\mathbf { h }\) s fallera id stan e \(x \mathrm {~m}\) ad \(k\) is a \(\mathbf { p }\) itie co tan.

1. Bys b in ra ra p p iate d fferen ial eq tin ,s how th t $$v ^ { 2 } = \frac { g } { k } \left( 1 - \mathrm { e } ^ { - 2 k x } \right)$$ [] It is \(\mathbf { N }\) it \(\mathbf { h } \mathrm { t } k = 0.01\). The speed of \(P \mathrm { w } \mathbf { h } \mathrm { n } x \mathbf { b } \mathrm {~cm}\) es larg ap \(\mathbf { o } \mathrm { ch } \mathrm { s } V \mathrm {~ms} ^ { - 1 }\).
2. 1. Fid \(V\) correct to 2 decimal places.
   2. Hen e fidw far \(P \mathbf { b }\) s fallenw \(\mathbf { b }\) n ts sp ed \(\mathrm { s } \frac { 1 } { 2 } V \mathrm {~ms} ^ { - 1 }\).
      \includegraphics[max width=\textwidth, alt={}]{1e5941a9-14eb-441c-8d39-fd62695446ac-08_319_908_255_580}
      Two in fo m smo h sp res \(A\) ad \(B \mathbf { 6 }\) eq \(l\) radili masses \(m\) ad \(2 m\) resp ctie ly. Sp re \(B\) is at rest \(\mathbf { n }\) a smo hb izn al sn face. Sp ere \(A\) is mi \(\mathbf { g } \mathbf { n }\) th sn face with sp ed \(u\) at an ag e \(\mathbf { b } \boldsymbol { B }\) to th lin 6 cen res \(6 A\) and \(B\) wh n it cb lid s with \(B\) (see \(\dot { \mathbf { d } }\) ag am). Th ce fficien

5 A \(\mathbf { p }\) rticle \(P \mathbf { 6 }\) mass \(m\) is attach d to e. a lig in k en ib e strig leg h \(a\). Tb o b re. th strig s attach d or fiæ \(\Phi \quad n O\).

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{1e5941a9-14eb-441c-8d39-fd62695446ac-10_314_768_397_651} Th \(\boldsymbol { p }\) rticle \(P\) mo sinab izo al circle with a co tan ag arsp ed \(\omega\) with th strig in lie d at \(\theta\) to \(\mathrm { b } \quad \mathrm { m }\) ard rtical th \(\mathrm { g } \quad O\) (see \(\dot { \mathrm { d } } \mathrm { ag } \mathrm { am }\) ). Sw that \(\omega ^ { 2 } = \frac { 2 g } { a }\).
2. Th \(\mathbf { p }\) rticle w has at restad stan e \(a\) rtically b low \(O\). It isth \(\mathrm { np } \dot { \mathrm { p } }\) ected b izb ally so th t it \(\mathbf { b } \mathbf { g }\) s to m in a rtical circle with cen re \(O\). Wh n th strig maks an ag e \(\boldsymbol { \theta }\), with th d \(\mathbb { w }\) ard rtical th \(\mathbf { g } O\), th ang arse \(\operatorname { d } P\) is \(\sqrt { \frac { 2 g } { a } }\). Tb strig first \(\mathbf { g }\) s slack when \(O P\) mak s ara g e \(\theta\) with b ard rtical th ob \(O\). Fid b le 6 co \(\theta\).

6 A \(\mathbf { p }\) rticle \(P\) is \(\mathrm { p } \dot { \mathbf { j } }\) ected with sp ed \(u\) at an ag \(\mathrm { e } \alpha\) ab tb \(\mathbf { b }\) izn al frm \(\mathrm { a } \dot { \mathbf { p } }\) n \(O \mathbf { n }\) ab izb al p ae ad mo s freely d rg av ty. Th \(\mathbf { b }\) izo al ad rtical d sp acemen s \(\mathbf { 6 } P\) frm \(O\) at a sb eq \(n\) time \(t\) are \(\mathbf { d } \mathbf { n }\) edy \(\quad x\) ad \(y\) resp ctie ly.

1. Derie the eq tirib the trajecto y \(P\) irt b fo m $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \sec ^ { 2 } \alpha$$
2. Tb g eatestb in \(6 P\) ab th p au is o edy \(H\). Wh \(\mathrm { n } P\) is ata \(\mathbf { b }\) ig \(6 \frac { 3 } { 4 } H\), it \(\mathbf { a }\) strac lled ab izn ald stan e \(d\). Gie it \(\mathbf { h } \tan \alpha = \Im\) id it erms \(6 H\),tb twœ sibed le \(\mathrm { s } \varnothing d\). If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n