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MathidiaMathematics Academy
Further Mathematics synthesis
Lesson 72 of 75
Easy practice

Further Applied synthesis

Objective. Select statistical, mechanical or discrete models.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Further Applied synthesis requires selecting the correct model from statistics, mechanics or discrete mathematics for a given real-world scenario, justifying the choice based on the structure of the problem, such as whether events are discrete or continuous, or whether motion involves collisions or circular paths. A synthesis question might present a scenario where a mechanics model (such as momentum conservation) feeds into a subsequent probability calculation, requiring both applied strands to be used in sequence. Recognising which branch of applied mathematics governs each stage of a multi-part problem is the key skill.

Worked example

A particle collides and rebounds with coefficient of restitution 0.5 from an initial speed of 8 m/s. Given the rebound speed is modelled as following a Poisson-adjusted count of impacts per minute at rate 3, find the rebound speed and the probability of exactly 2 impacts in a minute.

0.5832
Bar chart of the values 0.5, 8, 3, 2
  1. 1Rebound speed uses Newton's law of restitution directly: separation speed = e × approach speed = 0.5×8 = 4 m/s
  2. 2This mechanics calculation is independent of the Poisson process described
  3. 3For the Poisson part, λ=3 and r=2: P(X=2) = e^(−3)(3²)/2!
  4. 4Compute: e^(−3)≈0.0498, so P(X=2) ≈ 0.0498×9/2 ≈ 0.2240

Answer: Rebound speed = 4 m/s; P(X=2) ≈ 0.2240

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Common mistake. Students apply a mechanics formula to a problem that is actually asking for a probability distribution, or vice versa, because the scenario's wording resembles a familiar context rather than matching the actual mathematical structure. Always identify what quantity is being asked for before choosing a model.

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