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MathidiaMathematics Academy
A Level synthesis
Lesson 73 of 75
Standard practice

Mechanics synthesis

Objective. Connect forces, calculus and two-dimensional motion.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Mechanics synthesis connects forces, calculus and two-dimensional motion into a single solution, for example combining Newton's second law with vector resolution to find acceleration, then integrating that acceleration to track velocity and displacement over time, possibly alongside projectile or variable-acceleration ideas. A strong synthesis answer identifies which forces act, resolves them appropriately, applies F = ma to form equations, and then uses calculus methods if the acceleration is not constant, finishing with clear units and a sensible physical conclusion. For example, finding the resultant force on an object, then integrating its acceleration to obtain a velocity-time relationship, draws on both dynamics and calculus.

Worked example

A particle of mass 2 kg experiences a resultant force F = (6t) N along a line, starting from rest. Use mechanics synthesis to find its velocity at t = 3 seconds.

2 … 6
Illustrated card showing 2 … 6
  1. 1Newton's second law: a = F/m = 6t ÷ 2 = 3t
  2. 2Integrate: v = ∫3t dt = 1.5t² + c
  3. 3At t = 0, v = 0 (starts from rest), so c = 0
  4. 4At t = 3: v = 1.5(9) = 13.5 m/s

Answer: v = 13.5 m/s

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Common mistake. Students treat forces and calculus as separate topics, stopping after finding acceleration from F = ma instead of continuing to integrate for velocity or displacement when asked. Always check whether the question needs the full chain from force through to displacement.

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