Vectors and Motion in a Plane

Introduction

Motion in a plane, also known as two-dimensional motion, extends the concepts of motion from a straight line to scenarios where an object moves in a plane (like projectile motion or circular motion). This requires the use of vector quantities and mathematical tools to describe movement in two dimensions (typically x and y axes).

Understanding motion in a plane is crucial for describing a wide range of real-world phenomena, from the flight of a ball to the orbits of planets. We will explore how vectors are used to represent quantities like position, velocity, and acceleration in a plane, and how these concepts are applied to specific types of motion.

Scalars And Vectors

As discussed before, physical quantities can be broadly classified into scalars and vectors.

Position Vector And Displacement

In a plane (or higher dimensions), the position of an object is described by a position vector ($ \vec{r} $). This vector originates from the origin of the coordinate system (usually (0,0)) and points to the object's location.

• If an object is at coordinates $ (x, y) $, its position vector is $ \vec{r} = x \hat{i} + y \hat{j} $.
• Displacement: Displacement is the change in the position vector. If an object moves from an initial position $ \vec{r}_i $ to a final position $ \vec{r}_f $, its displacement $ \Delta \vec{r} $ is given by:

  $ \Delta \vec{r} = \vec{r}_f - \vec{r}_i $

• If $ \vec{r}_i = x_i \hat{i} + y_i \hat{j} $ and $ \vec{r}_f = x_f \hat{i} + y_f \hat{j} $, then

  $ \Delta \vec{r} = (x_f - x_i) \hat{i} + (y_f - y_i) \hat{j} $

  The components of the displacement vector are $ \Delta x = x_f - x_i $ and $ \Delta y = y_f - y_i $.

Equality Of Vectors

Two vectors are considered equal if and only if they have the same magnitude and the same direction. The position where the vector is drawn (its starting point or tail) does not affect its equality. A vector can be translated anywhere in space without changing its identity.

• If $ \vec{A} = A_x \hat{i} + A_y \hat{j} $ and $ \vec{B} = B_x \hat{i} + B_y \hat{j} $, then $ \vec{A} = \vec{B} $ if and only if $ A_x = B_x $ and $ A_y = B_y $.

Multiplication Of Vectors By Real Numbers

Multiplying a vector by a scalar (a real number) changes the magnitude of the vector and possibly its direction.

• Let $ \vec{A} $ be a vector and $ k $ be a real number (scalar).
• The vector $ k\vec{A} $ has a magnitude $ |k||\vec{A}| $.
• Direction:
  • If $ k > 0 $, the direction of $ k\vec{A} $ is the same as the direction of $ \vec{A} $.
  • If $ k < 0 $, the direction of $ k\vec{A} $ is opposite to the direction of $ \vec{A} $.
  • If $ k = 0 $, then $ k\vec{A} = \vec{0} $ (the zero vector).
• In terms of components: If $ \vec{A} = A_x \hat{i} + A_y \hat{j} $, then $ k\vec{A} = (kA_x) \hat{i} + (kA_y) \hat{j} $.

Example: If $ \vec{v} $ is a velocity vector, then $ 2\vec{v} $ is a velocity vector with twice the magnitude but the same direction. $ - \vec{v} $ is a vector with the same magnitude as $ \vec{v} $ but in the opposite direction.

Addition And Subtraction Of Vectors — Graphical Method

As discussed previously in the context of motion in a straight line, graphical methods can be used for vector addition and subtraction in a plane.

• Triangle Law: To add A and B, place the tail of B at the head of A. The resultant R is the vector from the tail of A to the head of B.
• Parallelogram Law: Draw A and B from a common origin. Complete the parallelogram. The resultant R is the diagonal starting from the common origin.
• Subtraction ($ \mathbf{A} - \mathbf{B} $): This is equivalent to adding $ \mathbf{A} + (-\mathbf{B}) $. Draw vector B, then reverse its direction to get $ -\mathbf{B} $, and then apply the addition law.

These graphical methods are useful for visualizing vector operations but may lack precision for calculations.

Resolution Of Vectors

Resolution of a vector is the process of breaking down a vector into two or more component vectors that add up to the original vector. The most common way to resolve a vector is into components along the mutually perpendicular x and y axes (in 2D) or x, y, and z axes (in 3D).

Consider a vector $ \vec{A} $ in a plane, making an angle $ \theta $ with the positive x-axis.

• Components:
  • The component of $ \vec{A} $ along the x-axis is $ A_x = |\vec{A}| \cos \theta $.
  • The component of $ \vec{A} $ along the y-axis is $ A_y = |\vec{A}| \sin \theta $.
• Vector Representation using Components: The vector $ \vec{A} $ can be written in terms of unit vectors $ \hat{i} $ (along x-axis) and $ \hat{j} $ (along y-axis) as:

  $ \vec{A} = A_x \hat{i} + A_y \hat{j} $

• Magnitude of the Vector: From the Pythagorean theorem, the magnitude of $ \vec{A} $ is:

  $ |\vec{A}| = \sqrt{A_x^2 + A_y^2} $

• Direction of the Vector: The angle $ \theta $ can be found using trigonometry:

  $ \tan \theta = \frac{A_y}{A_x} $

Resolution is a fundamental step in the analytical method of vector addition and in solving problems involving forces or velocities in multiple directions.

Vector Addition – Analytical Method

The analytical method for adding vectors uses the resolution of vectors into their components.

Steps to add vectors $ \vec{A} $, $ \vec{B} $, etc., in a plane:

1. Resolve each vector into its x and y components:
   • $ \vec{A} = A_x \hat{i} + A_y \hat{j} $
   • $ \vec{B} = B_x \hat{i} + B_y \hat{j} $
2. Add the corresponding components: The resultant vector $ \vec{R} = \vec{A} + \vec{B} $ is found by adding the x-components together and the y-components together.

   $ \vec{R} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j} $

   Let $ R_x = A_x + B_x $ and $ R_y = A_y + B_y $. So, $ \vec{R} = R_x \hat{i} + R_y \hat{j} $.
3. Calculate the magnitude of the resultant vector:

   $ R = |\vec{R}| = \sqrt{R_x^2 + R_y^2} $

4. Calculate the direction of the resultant vector: The angle $ \theta $ the resultant vector makes with the positive x-axis is given by:

   $ \tan \theta = \frac{R_y}{R_x} $

   Ensure the angle is in the correct quadrant based on the signs of $ R_x $ and $ R_y $.

This analytical method is precise and can be extended to adding any number of vectors.

Motion In A Plane

Motion in a plane involves describing the position, velocity, and acceleration of an object that is moving in two dimensions.

Position Vector And Displacement

As mentioned earlier, in a plane, the position is described by a position vector $ \vec{r}(t) = x(t) \hat{i} + y(t) \hat{j} $, where $ x(t) $ and $ y(t) $ are the coordinates at time $ t $. Displacement $ \Delta \vec{r} $ is the change in this position vector: $ \Delta \vec{r} = \vec{r}_f - \vec{r}_i $. The magnitude of displacement is the straight-line distance between the initial and final points, while its direction points from the initial to the final point.

Example: A particle moves from (1, 2) to (4, 6).

• $ \vec{r}_i = 1 \hat{i} + 2 \hat{j} $
• $ \vec{r}_f = 4 \hat{i} + 6 \hat{j} $
• $ \Delta \vec{r} = (4-1) \hat{i} + (6-2) \hat{j} = 3 \hat{i} + 4 \hat{j} $.
• Magnitude of displacement = $ \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5 $ units.
• Direction of displacement: $ \tan \theta = 4/3 $, so $ \theta = \arctan(4/3) $.

Motion In A Plane With Constant Acceleration

If an object moves in a plane with constant acceleration, its motion can be analyzed by considering the x and y components of motion independently. This is because acceleration is a vector, and if it's constant, both its magnitude and direction are constant. Therefore, the x-component of acceleration ($ a_x $) is constant, and the y-component of acceleration ($ a_y $) is constant.

• Velocity in a plane:

  $ \vec{v}(t) = \vec{v}_0 + \vec{a}t $

  This vector equation can be broken down into component equations:

  $ v_x(t) = v_{0x} + a_x t $

  $ v_y(t) = v_{0y} + a_y t $

  where $ \vec{v}_0 = v_{0x} \hat{i} + v_{0y} \hat{j} $ is the initial velocity and $ \vec{v}(t) = v_x(t) \hat{i} + v_y(t) \hat{j} $ is the velocity at time $ t $.
• Position in a plane:

  $ \vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2}\vec{a}t^2 $

  This vector equation also breaks down into component equations:

  $ x(t) = x_0 + v_{0x} t + \frac{1}{2}a_x t^2 $

  $ y(t) = y_0 + v_{0y} t + \frac{1}{2}a_y t^2 $

  where $ \vec{r}_0 = x_0 \hat{i} + y_0 \hat{j} $ is the initial position.

These equations are very similar to the one-dimensional kinematic equations, but applied independently to each component of motion.

Relative Velocity In Two Dimensions

The concept of relative velocity extends to two dimensions. If we have three frames of reference: S (e.g., ground), S' (moving relative to S), and particle P (moving relative to S'), then the velocity of P relative to S ($ \vec{v}_{PG} $) can be found using the velocities relative to each frame.

The general principle is:

$ \vec{v}_{PG} = \vec{v}_{PS'} + \vec{v}_{S'G} $

Where:

• $ \vec{v}_{PG} $ is the velocity of particle P relative to frame S (ground).
• $ \vec{v}_{PS'} $ is the velocity of particle P relative to frame S'.
• $ \vec{v}_{S'G} $ is the velocity of frame S' relative to frame S (ground).

This is a vector addition. If frame S' is moving with velocity $ \vec{v}_B $ relative to S, and particle P has velocity $ \vec{v}_A $ relative to S', then the velocity of P relative to S is:

$ \vec{v}_{PG} = \vec{v}_A + \vec{v}_B $

Example: A boat moving across a river. The boat's velocity relative to the water ($ \vec{v}_{boat, water} $) and the water's velocity relative to the ground ($ \vec{v}_{water, ground} $) add up to the boat's velocity relative to the ground ($ \vec{v}_{boat, ground} $).

Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration due to gravity (ignoring air resistance). This motion is confined to a plane.

• Assumptions:
  • Acceleration due to gravity ($ \vec{g} $) is constant and acts vertically downwards.
  • Air resistance is negligible.
• Decomposition of Motion: Projectile motion is analyzed by treating the horizontal and vertical components of motion independently.
  • Horizontal Motion: Since there is no horizontal acceleration ($ a_x = 0 $), the horizontal velocity $ v_x $ remains constant throughout the flight. The horizontal displacement $ x $ is given by $ x = v_{0x} t $, where $ v_{0x} $ is the initial horizontal velocity.
  • Vertical Motion: The vertical motion is under constant acceleration due to gravity ($ a_y = -g $, assuming upward direction is positive). The vertical component of initial velocity is $ v_{0y} $. The vertical position $ y $ and vertical velocity $ v_y $ at time $ t $ are given by the kinematic equations:

    $ y(t) = v_{0y}t - \frac{1}{2}gt^2 $

    $ v_y(t) = v_{0y} - gt $

• Trajectory: The path traced by a projectile is a parabola. The equation of the trajectory can be found by eliminating time $ t $ from the x and y equations: $ y = x \tan \theta_0 - \frac{g x^2}{2 v_0^2 \cos^2 \theta_0} $, where $ v_0 $ is the initial speed and $ \theta_0 $ is the launch angle.
• Key Quantities: Time of flight, maximum height, range (horizontal distance covered).

Uniform Circular Motion

Uniform Circular Motion (UCM) is motion in a circular path at a constant speed. Although the speed is constant, the velocity is not, because the direction of motion is continuously changing.

• Velocity: The velocity vector is always tangent to the circular path at the object's position.
• Acceleration: There is a constant acceleration directed towards the center of the circle, called centripetal acceleration ($ a_c $). Its magnitude is $ a_c = \frac{v^2}{r} $, where $ v $ is the constant speed and $ r $ is the radius of the circle.
• Force: A centripetal force ($ F_c = ma_c = \frac{mv^2}{r} $) is required to maintain this circular motion, acting towards the center of the circle.
• Period (T): The time taken for one complete revolution is $ T = \frac{2\pi r}{v} $.
• Frequency (f): The number of revolutions per second is $ f = \frac{1}{T} $.
• Angular Velocity ($ \omega $): The rate of change of angle, related to linear speed by $ v = \omega r $.

UCM is an important example of motion where acceleration exists even though speed is constant.