Vector Addition in a Plane

When two vectors A and B act at an angle θ, their resultant R can be found using the parallelogram method:

  • Magnitude of R: \[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \] (Law of Cosines)
  • Direction of R (angle α with vector A): \[ \tan \alpha = \frac{B \sin \theta}{A + B \cos \theta} \] The direction can also be found using the Law of Sines: \[ \frac{R}{\sin \theta} = \frac{A}{\sin \beta} = \frac{B}{\sin \alpha} \]

Position Vector and Displacement

An object’s position in a plane is described by a position vector:

\[ \mathbf{r} = x \hat{i} + y \hat{j} \]

Displacement (Δ𝐫) is the change in position:

\[ \Delta \mathbf{r} = (x’ – x) \hat{i} + (y’ – y) \hat{j} = \Delta x \hat{i} + \Delta y \hat{j} \]

Velocity in Two Dimensions

  • Average velocity: \[ \overline{\mathbf{v}} = \frac{\Delta \mathbf{r}}{\Delta t} = \overline{v}_x \hat{i} + \overline{v}_y \hat{j} \]
  • Instantaneous velocity (direction is tangent to the path): \[ \mathbf{v} = \frac{d\mathbf{r}}{dt} = v_x \hat{i} + v_y \hat{j} \] where \( v_x = \frac{dx}{dt} \), \( v_y = \frac{dy}{dt} \)
  • Speed (magnitude of velocity): \[ v = \sqrt{v_x^2 + v_y^2} \]
  • Direction of velocity: \[ \theta = \tan^{-1} \left( \frac{v_y}{v_x} \right) \]

Acceleration in Two Dimensions

  • Average acceleration: \[ \overline{\mathbf{a}} = \frac{\Delta \mathbf{v}}{\Delta t} = \frac{\Delta v_x}{\Delta t} \hat{i} + \frac{\Delta v_y}{\Delta t} \hat{j} \]
  • Instantaneous acceleration: \[ \mathbf{a} = \frac{d\mathbf{v}}{dt} = a_x \hat{i} + a_y \hat{j} \] where \( a_x = \frac{dv_x}{dt} \), \( a_y = \frac{dv_y}{dt} \)

Motion with Constant Acceleration

If acceleration a is constant, velocity and position can be calculated using:

  • Velocity at time t: \[ \mathbf{v} = \mathbf{v}_0 + \mathbf{a}t \] (components: \( v_x = v_{0x} + a_x t \), \( v_y = v_{0y} + a_y t \))

Example Problems

  1. Resultant Velocity: A boat moving north at 25 km/h and a current of 10 km/h at 60° east of south: \[ R = \sqrt{25^2 + 10^2 + 2 \cdot 25 \cdot 10 \cdot \cos 120^\circ} \approx 22 \, \text{km/h} \] Direction angle: φ ≈ 23.4°.
  2. Velocity from Position: For \( \mathbf{r} = 3.0t \hat{i} + 2.0t^2 \hat{j} + 5.0 \hat{k} \): \[ \mathbf{v} = 3.0 \hat{i} + 4.0t \hat{j}, \quad \mathbf{a} = 4.0 \hat{j} \, \text{m/s}^2 \] At t=1s: \( v = 5.0 \, \text{m/s} \), θ = 53° with x-axis.

Key NEET Topics

  • Vector addition using Law of Cosines and Law of Sines.
  • Finding velocity and acceleration from position-time equations.
  • Direction of velocity as the tangent to the path.
  • Resolving vectors into components (\( v_x = v \cos \theta, v_y = v \sin \theta \)).
  • Constant acceleration equations in 2D.