Standard +0.3 This is a standard two-part collision problem requiring conservation of momentum and Newton's restitution law. Part (i) involves straightforward algebraic manipulation to find e in terms of α. Part (ii) requires recognizing that 0 ≤ e ≤ 1, leading to a simple inequality. While it requires multiple steps and careful algebra, it follows a well-established template for A-level mechanics collision problems with no novel insight needed.
Two smooth spheres \(A\) and \(B\), of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere \(A\) has mass \(m\) and speed \(u\) and sphere \(B\) has mass \(\alpha m\) and speed \(\frac{1}{4}u\). The spheres collide and \(A\) is brought to rest by the collision. Find the coefficient of restitution in terms of \(\alpha\). [6]
Deduce that \(\alpha \geqslant 2\). [2]
Use conservation of momentum: mu + ¼α mu = α mvB M1 A1
Use Newton’s law of restitution: – vB = – e(u – ¼u) [vB = ¾eu] M1 A1
Eliminate vB to find e (A.E.F.): e = (1 + ¼α)/¾α or (4 + α)/3α M1 A1
Answer
Marks
Use e ≤ 1 to find inequality for α: 4 + α ≤ 3α so α ≥ 2 A.G. M1 A1
6
2
[8]
Question 3:
3 | Use conservation of momentum: mu + ¼α mu = α mvB M1 A1
Use Newton’s law of restitution: – vB = – e(u – ¼u) [vB = ¾eu] M1 A1
Eliminate vB to find e (A.E.F.): e = (1 + ¼α)/¾α or (4 + α)/3α M1 A1
Use e ≤ 1 to find inequality for α: 4 + α ≤ 3α so α ≥ 2 A.G. M1 A1 | 6
2 | [8]
Two smooth spheres $A$ and $B$, of equal radius, are moving in the same direction in the same straight line on a smooth horizontal table. Sphere $A$ has mass $m$ and speed $u$ and sphere $B$ has mass $\alpha m$ and speed $\frac{1}{4}u$. The spheres collide and $A$ is brought to rest by the collision. Find the coefficient of restitution in terms of $\alpha$. [6]
Deduce that $\alpha \geqslant 2$. [2]
\hfill \mbox{\textit{CAIE FP2 2010 Q3 [8]}}