A2 October 2020 Q3

EdexcelCurrent spec14 marksCollisions: 2 Spheres Direct

3. Two particles, \(A\) and \(B\), have masses \(3m\) and \(4m\) respectively. The particles are moving in the same direction along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision the speed of \(A\) is \(2u\) and the speed of \(B\) is \(u\).

The coefficient of restitution between \(A\) and \(B\) is \(e\).

(a) Show that the direction of motion of each of the particles is unchanged by the collision. (8)

After the collision with \(A\), particle \(B\) collides directly with a third particle, \(C\), of mass \(2m\), which is at rest on the surface.

The coefficient of restitution between \(B\) and \(C\) is also \(e\).

(b) Show that there will be a second collision between \(A\) and \(B\). (6)

Diagram: A (3m) moving right at 2u, B (4m) moving right at u, C (2m) at rest; after the first collision A moves right at v and B at w; after the second collision B moves right at x and C at y — Mark scheme (a)

Scheme	Marks	AO
Taking left to right as positive,
CLM:	M1	3.1a
\(6mu + 4mu\ (= 10mu) = 3mv + 4mw\) \((10u = 3v + 4w)\)	A1	1.1b
Impact Law:	M1	3.4
\(w - v = e(2u - u)\ (= eu)\)	A1	1.1b
Solve for \(v\) or \(w\)	M1	2.1
\(w = \dfrac{u}{7}(10 + 3e)\)	A1	1.1b
\(v = \dfrac{u}{7}(10 - 4e)\)	A1	1.1b
\(0 \leqslant e \leqslant 1\ \ \Rightarrow\ \ 10 + 3e \gt 0\) and \(10 - 4e \gt 0\) hence both particles still travelling in the original direction. *	A1*	2.2a
	(8)

Mark scheme (b)

Scheme	Marks	AO
CLM: \(4mw = 4mx + 2my\) \((2w = 2x + y)\)	M1	3.1a
Impact: \(y - x = ew\)	M1	3.4
\(\Rightarrow w(2 - e) = 3x\), \(x = \dfrac{u}{21}(10 + 3e)(2 - e)\)	M1	1.1b
Consider \(v - x\) i.e. \(\dfrac{u}{7}(10 - 4e) - \dfrac{u}{21}(10 + 3e)(2 - e)\) \((3e^2 - 8e + 10)\)	M1	2.1
Show that \(v - x \gt 0\ \forall e\)	M1	1.1b
Complete correct argument and conclusion *	A1*	2.2a
	(6)
	(14 marks)

Notes

M1: All terms required. Condone sign errors

M1: Correct use of impact law. Condone sign errors

M1: Use their correctly formed equations to find velocity of \(B\) (\(x\))

M1: Form relevant difference for a second collision

M1: Complete correct method (e.g. differentiation or completing the square or discriminant) to determine when inequality is true

A1*: Reach correct conclusion from correct work.