Momentum

• State “+”, draw before and after, sign every velocity, read the direction off the answer’s sign.
• \(p = mv\), signed. \(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\); coalesced, \((m_1 + m_2)V\). Unknowns drawn positive; a negative answer is a direction, said in words.

• 4.3.1 I can use — use the definition of linear momentum and show understanding of its vector nature
• 4.3.2 I can use — use conservation of linear momentum to solve problems that may be modelled as the direct impact of two bodies

Accuracy for this chapter. Most momentum answers come out exact: leave them exact (\(\tfrac{2}{3}\) kg, \(\tfrac{10}{3}\ \text{m s}^{-1}\)) or give a non-exact decimal to 3 significant figures (\(0.667\) kg, \(3.33\ \text{m s}^{-1}\)). A velocity is not finished until its direction is stated in words: “\(v = -1\)” becomes “\(1\ \text{m s}^{-1}\) in the opposite direction to A’s original motion”. Write the conservation equation, with the signs, before any number: an unsupported answer earns nothing.

• “B moves at \(3\ \text{m s}^{-1}\), so I write \(6(3)\) for its momentum.” Correct A body moving the other way has negative velocity. State the positive direction first; if B moves against it, its velocity is \(-3\) and its momentum \(6(-3) = -18\). Leaving the sign out turns \(20 - 18 = 2\) into \(38\).
• “They stick together, so \(m_1u_1 + m_2u_2 = m_1V\).” Correct Coalesced bodies move as one body of mass \(m_1 + m_2\): \(m_1u_1 + m_2u_2 = (m_1 + m_2)V\).
• “\(v = -1\), so the speed is \(-1\ \text{m s}^{-1}\).” Correct The sign is the direction. The speed is \(1\ \text{m s}^{-1}\), and the body moves in the negative direction; say so in words.
• “The ball hits the wall at \(15\) and leaves at \(10\), so its momentum changes by \(0.2(15 - 10) = 1\).” Correct After the rebound its velocity is \(-10\). The change is \(0.2(-10) - 0.2(15) = -5\): a magnitude of \(5\ \text{kg m s}^{-1}\), directed away from the wall.
• “I wrote A’s velocity as \(-v\), got \(v = -1\), so A moves back.” Correct If the unknown was written as \(-v\), its velocity is \(-v = +1\): A moves forwards. A minus sign written in advance is legitimate only if you then read the answer through it. The safe habit is to draw every unknown in the positive direction and let the algebra give the sign.
• “Momentum is conserved, so each body keeps its own momentum.” Correct Only the total of the two bodies is conserved. What one gains, the other loses.
• “Momentum \(= \tfrac{1}{2}mv^2\).” Repair Momentum is \(mv\), a vector. \(\tfrac{1}{2}mv^2\) is kinetic energy, a scalar, defined in chapter 22 and not used here.
• “\(4(5) + 6(3) = 38\)” for two bodies moving towards each other. Repair State a positive direction. B moves against it, so its velocity is \(-3\) and the total is \(20 - 18 = 2\).
• “\(v = -1\), so the speed is \(-1\ \text{m s}^{-1}\).” Repair The sign is the direction. The speed is \(1\ \text{m s}^{-1}\) and the body moves in the negative direction.
• “\(v = -1\) must be an error; I will check my working.” Repair A negative answer is a result, not a mistake: the body has rebounded. Say so in words.
• “They coalesce, so \(m_1u_1 + m_2u_2 = m_1V\).” Repair After coalescing, the moving mass is \(m_1 + m_2\): \(m_1u_1 + m_2u_2 = (m_1 + m_2)V\).
• “The ball’s change in momentum is \(0.2(15 - 10) = 1\).” Repair After a rebound the final velocity is \(-10\). The change is \(0.2(-10 - 15) = -5\): \(5\ \text{kg m s}^{-1}\) away from the wall.
• “A bounces back, so I call its velocity \(-v\)”, and then the answer comes out negative as well. Repair Draw every unknown in the positive direction and let the algebra give the sign. If A’s velocity was written as \(-v\) and the answer is \(v = -1\), then A’s velocity is \(-v = +1\): it moves forwards, not back. Reading the sign of \(v\) as the direction, when the working already assumed one, sends the body the wrong way.
• “Momentum is conserved, so each body keeps its own momentum.” Repair Only the total is conserved. One body gains exactly what the other loses.
• An answer in which the body behind ends up moving faster than the body in front, in the same direction. Repair It would have to pass through the other body. Recheck the signs; the physical check is step 7 of the method card.
• “The unit of momentum is \(\text{kg m s}^{-2}\).” Repair Mass times velocity: \(\text{kg m s}^{-1}\), or equivalently \(\text{N s}\). \(\text{kg m s}^{-2}\) is the newton, a unit of force.
• Using P’s speed as it passes A as its speed at the impact, when P is decelerating between the two. Repair Find the speed at the impact first, with a constant-acceleration formula, then use it in the conservation equation.

• Read the command word before you decide how much to write. This syllabus uses eleven: calculate, describe, determine, evaluate, explain, identify, justify, show (that), sketch, state and verify. Show that and verify give you the answer and mark the route to it, so every step must be visible and the argument must run forwards from what is given, never backwards from the result. Sketch means a simple freehand drawing showing the key features, taking care over proportions; it is not a plot. Determine means establish with certainty; justify means support a case with evidence or argument. Find, solve, express and hence are ordinary question wording; hence means the previous part is the intended route.
• Interleave with the chapters that use this one. Chapter 22 defines kinetic energy, a scalar: when you reach it, re-answer the first item of the mistake clinic and retrieval question 17, so that momentum, a vector, stays distinct from it. Chapter 21 connects force and motion: re-read the third-law reason in section B alongside it. Recalling a method inside a new problem is worth more than another pass over this chapter on its own: later chapters use these methods without re-teaching them, and the syllabus says an individual examination question may involve ideas and methods from more than one section of the content for that paper, so nothing here is ever finished with.

• Cambridge International AS & A Level Mathematics 9709 has six components, and a candidate takes two of them for the AS Level and four for the A Level. This chapter is Mechanics content, examined in Paper 4. Paper 4 (Mechanics) is 40% of an AS Level that includes it and 20% of an A Level that includes it. It assumes knowledge of the algebraic methods in the Paper 1 content. An A Level route that includes Paper 4 takes Papers 1, 3, 4 and 5; Paper 4 cannot be combined with Paper 6. Every paper is a written examination of compulsory structured questions, answered on the question paper, with MF19 (the list of formulae and statistical tables) supplied. Examinations are available in the June and November series, and in March in India.
• Across the whole qualification the assessment objectives are weighted AO1 55% (knowledge and understanding: concepts, terminology, notation and accurate manipulative technique) and AO2 45% (application and communication: choosing the procedure, combining techniques to solve problems, and presenting the work clearly and logically) at AS Level, and AO1 52%, AO2 48% at A Level. AS candidates are graded a–e; A Level candidates A*–E.
• Three forms: find an unknown velocity or mass after a direct impact; show that a direction is reversed, where the conclusion has to be read off the sign of the answer and said in words; and a multi-part problem in which a constant-acceleration stage leads into an impact and a second stage follows it. The reasoning lives in the signed conservation equation, so write it in full.
• MF19 prints only the four constant-acceleration formulae for Mechanics. Momentum \(= mv\), conservation of momentum and the coalescence equation must all be known. See the MF19 card.
• A missing minus sign on a body moving the other way; a coalesced mass written as one body’s mass; a negative answer left without its direction in words. Answers are often exact (\(\tfrac{2}{3}\) kg); otherwise give 3 significant figures, and use \(g = 10\ \text{m s}^{-2}\) if a stage needs it.
• Read the command word before you decide how much to write. This syllabus uses eleven: calculate, describe, determine, evaluate, explain, identify, justify, show (that), sketch, state and verify. Show that and verify give you the answer and mark the route to it, so every step must be visible and the argument must run forwards from what is given, never backwards from the result. Sketch means a simple freehand drawing showing the key features, taking care over proportions; it is not a plot. Determine means establish with certainty; justify means support a case with evidence or argument. Find, solve, express and hence are ordinary question wording; hence means the previous part is the intended route.