A-Level Mathematics May/June 2025 Q2: Two particles P and Q, of masses 0.2kg and 0.1kg respectively, are free to move in a st…

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  accepted ✓

  Let the direction of motion of P be the positive direction. Both particles are moving in the same direction.

  1. Conservation of Linear Momentum

  By the principle of conservation of linear momentum: Total momentum before collision = Total momentum after collision $m_P u_P + m_Q u_Q = (m_P + m_Q)v$

  Substituting the given values: $(0.2)(5) + (0.1)(2) = (0.2 + 0.1)v$ $1 + 0.2 = 0.3v$ $1.2 = 0.3v$ $v = \frac{1.2}{0.3} = 4$ ms⁻¹

  The common velocity of the coalesced particles is 4 ms⁻¹.

  2. Kinetic Energy Loss

  Kinetic Energy (KE) is given by the formula $KE = \frac{1}{2}mv^2$.

  Total initial KE = KE of P + KE of Q $KE_{initial} = \frac{1}{2}(0.2)(5^2) + \frac{1}{2}(0.1)(2^2)$ $KE_{initial} = \frac{1}{2}(0.2)(25) + \frac{1}{2}(0.1)(4)$ $KE_{initial} = 2.5 + 0.2 = 2.7$ J

  Total final KE = KE of combined particle $KE_{final} = \frac{1}{2}(0.2 + 0.1)(4^2)$ $KE_{final} = \frac{1}{2}(0.3)(16)$ $KE_{final} = 2.4$ J

  Loss in KE = $KE_{initial} - KE_{final}$ Loss in KE = $2.7 - 2.4 = 0.3$ J

  The kinetic energy lost during the collision is $0.3$ J.

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  How the marks are awarded

  • M1 — For applying the principle of conservation of linear momentum. The equation $(0.2)(5) + (0.1)(2) = (0.2 + 0.1)v$ shows a correct attempt with the right number of terms, even if a velocity sign had been incorrect.
  • A1 — For correctly calculating the common velocity after collision as $v = 4$ ms⁻¹.
  • DM1 — For a correct method to find the change in kinetic energy. This involves calculating the total initial KE and the total final KE and finding their difference, as shown in the calculation: $( \frac{1}{2}(0.2)(5^2) + \frac{1}{2}(0.1)(2^2) ) - ( \frac{1}{2}(0.3)(4^2) )$. This mark is dependent on the first M1.
  • A1 — For obtaining the final correct answer of $0.3$ J for the kinetic energy lost.

  Common mistakes

  • Misinterpreting the setup and assigning a negative velocity to particle Q, leading to an incorrect momentum equation like $0.2(5) + 0.1(-2) = 0.3v$. This results in an incorrect final velocity and loses all subsequent accuracy marks.
  • Calculating the change in KE by finding the change in velocity first and using an incorrect formula like $\frac{1}{2}m(\Delta v)^2$. The loss of KE must be calculated as the difference between the total final KE and the total initial KE.
  • Making arithmetic errors when squaring velocities or multiplying by fractions, for example calculating $\frac{1}{2}(0.3)(4^2)$ as $0.15 \times 8 = 1.2$ instead of $0.15 \times 16 = 2.4$.
  • Calculating the final KE as two separate terms, e.g., $\frac{1}{2}(0.2)(4^2) + \frac{1}{2}(0.1)(4^2)$, instead of using the combined mass $\frac{1}{2}(0.3)(4^2)$. While this gives the same result, it can lead to errors and shows a weaker understanding of the 'coalesce' concept.

  Examiner tip: Always begin by applying the conservation of linear momentum to find the unknown velocity after a collision before calculating any change in kinetic energy.

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