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 =3?.