Using Calculus in 2D (OCR A Level Maths: Mechanics)

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

  

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

(a)  Find an expression for the position vector of the ship after  seconds.

(b)  Find the magnitude of the acceleration when .