Challenging +1.8 This is an oblique collision problem requiring resolution of velocities along and perpendicular to the line of centres, application of conservation of momentum and Newton's restitution law in the impact direction, then recombination of components. While mechanically involved with multiple steps and careful bookkeeping of components, it follows a standard procedure taught in Further Maths mechanics without requiring novel insight or particularly subtle reasoning.
\includegraphics{figure_8}
A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]
Components of velocity ⊥ to AB are 5 and 4 ms –1 .
Speed A = 32 +52 = 34 or 5.83 ms –1 ,
Answer
Marks
Speed B = 72 +42 = 65 or 8.06 ms –1 (OE)
Any signs on RHS for A1
Consistent signs on RHS for
A1
(both √ on x-components)
Both
either
Both answers, correct to 3SF if
Answer
Marks
necessary
B1
M1A1√
M1A1√
A1
B1
M1
A1 [9]
Question 8:
8 | –1
Components of speed || to AB are 12 and 3 ms .
CLM: 2 × 12 – 3 × 3 = 2vA + 3vB
2
NEL: − (12+3)=v −v
3 A B
vA = –3, vB = 7
Components of velocity ⊥ to AB are 5 and 4 ms –1 .
Speed A = 32 +52 = 34 or 5.83 ms –1 ,
Speed B = 72 +42 = 65 or 8.06 ms –1 (OE) | Any signs on RHS for A1
Consistent signs on RHS for
A1
(both √ on x-components)
Both
either
Both answers, correct to 3SF if
necessary | B1
M1A1√
M1A1√
A1
B1
M1
A1 [9]
\includegraphics{figure_8}
A smooth sphere with centre $A$ and of mass 2 kg, moving at 13 m s$^{-1}$ on a smooth horizontal plane, strikes a smooth sphere with centre $B$ and of mass 3 kg moving at 5 m s$^{-1}$ on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles $\tan^{-1}\left(\frac{2}{13}\right)$ to $\overrightarrow{AB}$ and $\tan^{-1}\left(\frac{4}{3}\right)$ to $\overrightarrow{BA}$ respectively (see diagram). Given that the coefficient of restitution is $\frac{2}{3}$, find the speeds of the spheres after impact. [9]
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2014 Q8 [9]}}