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AQA A Level Maths: Mechanics复习笔记2.4.1 Using Calculus in 2D

Category: A-level课程, 教材笔记, 福利干货 Date: 2022年7月18日 下午12:10

Using Calculus in 2D

 

How can I use vectors in 2D Kinematics?

  • It is important you understand?Calculus in 1D?first
  • Also recall the differences between key?vector?and?scalar?quantities
  • Displacement?- from starting point
  • Distance?– from start/origin/another particle (all could be different)
  • Speed?and?Velocity
  • acceleration?and?magnitude?of?acceleration/deceleration
  • time?is a scalar quantity

How do I use calculus with vectors in 2D?

  • To?differentiate?a vector:
    • Differentiate both (i?and?j) components of the vector

     

  • To integrate?a vector:
    • Integrate both (i?and?j) components of the vector
    • There will be a constant of integration,?c, which will be a vector made up of two values (an?i-component constant and a?j-component constant)
    • Find?c?by substituting in any known vectors, in the same way as you would if it were in 1D
      • Find the separate components of the vector?c?by equating the?i-components and the?j-components separately

       

     

How does calculus link to kinematics?

  • In 2D we normally use the vector?r?to represent displacement instead of s
  • Just like in 1D:
    • Differentiate?displacement to get velocity and velocity to get acceleration
    • Integrate?acceleration to get velocity and velocity to get displacement

    2-4-1-using-calculus-in-2d-diagram-1

Harder problems involving vectors, calculus, position vectors and F=ma

  • Harder problems could involve:
    • using Newton’s Second Law (N2L)?F = ma
    • two vectors being equal requiring both?i?and?j?components to be equal
    • exponentials, logarithms, trigonometric functions (not just polynomials)

     

  • suvat equations (in vector form) would only be involved if?acceleration is constant
  • You may have to find the?magnitude?of the vectors
    • The magnitude of the?velocity?is the?speed
    • The magnitude of the?displacement?is the?distance?from the?starting point

     

  • The position vector of a particle is?r = r0?+ s
    • r0?is the?initial position?of the particle
    • s?is the?displacement?of the particle from its initial position

     

Worked Example

2.4.1_WE_Using-Calculus-in-2D_1

2-4-1-using-calculus-in-2d-example-solution-a

2-4-1-using-calculus-in-2d-example-solution-b

Exam Tip

  • If the question refers to the direction that the particle is travelling, then you would use the velocity. If the?direction?of?motion?is asked for then it should be clear from the question whether they want an angle, bearing or vector which it is parallel to.

 

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