Using Calculus 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

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 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

2.4.1_WE_Using Calculus in 2D_1

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

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

(b) Find the magnitude of the acceleration when $t = 3$ .

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

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.

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