M4 June 2006 Q6

EdexcelOld spec14 marksCollisions: 2 Spheres Oblique

Figure 2: spheres A (mass m) and B (mass 2m) in contact with centres on line L, each moving with speed U at 45 degrees to L towards the other — Figure 2

Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of 45\(^\circ\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\tfrac{1}{2}\).

(a) Find the magnitude of the impulse which acts on \(A\) in the collision. (9)

Figure 3: line L parallel to a vertical wall at distance d — Figure 3

The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.

(b) Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall. (5)

Before and after components: A (m) has U/root 2 along and across the line of centres before, v2 along and U/root 2 across after, impulse I; B (2m) has U/root 2 along and across before, v1 along and U/root 2 across after — Mark scheme (a)

Scheme	Marks
Form: \(\ I = m\left(v_2 + \dfrac{U}{\sqrt{2}}\right)\)	M1 A1
CLM\((\uparrow)\colon\ \dfrac{2mU}{\sqrt{2}} - \dfrac{mU}{\sqrt{2}} = 2mv_1 + mv_2\)
\(\dfrac{U}{\sqrt{2}} = 2v_1 + v_2\qquad (1)\)	M1 A1
NIL: \(\ e\dfrac{2U}{\sqrt{2}} = \dfrac{U}{\sqrt{2}} = -v_1 + v_2\qquad (2)\)	M1 A1
\(\Rightarrow \dfrac{3U}{\sqrt{2}} = 3v_2\)	M1 A1
\(\Rightarrow I = m\left(\dfrac{U}{\sqrt{2}} + \dfrac{U}{\sqrt{2}}\right)\)
\(= mU\sqrt{2}\)	A1
	(9)

Mark scheme (b)

Scheme	Marks
\(v_2 - v_1 = \dfrac{U}{\sqrt{2}}\qquad\) (Separation speed)	M1
time to wall \(= \dfrac{d}{U/\sqrt{2}} = \dfrac{d\sqrt{2}}{U}\)	M1 A1
\(\therefore\) Separation \(= \dfrac{d\sqrt{2}}{U} \times \dfrac{U}{\sqrt{2}} = d\)	M1 A1
	(5)
	(14 marks)

Notes