Motion in Two Dimensions

Understanding vectors, projectile motion, and relative motion in the plane

Introduction to Motion in Two Dimensions

Motion in two dimensions involves movement that can't be described using a single coordinate. Instead, we need to consider how an object moves along two perpendicular axes (typically x and y). This requires the use of vectors to represent quantities like position, velocity, and acceleration. From thrown basketballs to orbiting satellites, two-dimensional motion is everywhere around us.

In this section:
  1. Vectors in Two Dimensions
  2. Position and Displacement
  3. Velocity in Two Dimensions
  4. Acceleration in Two Dimensions
  5. Projectile Motion
  6. Relative Motion

Vectors in Two Dimensions

A vector is a quantity that has both magnitude (size) and direction. In two dimensions, vectors can be represented by:

  • An arrow with length proportional to its magnitude and pointing in the direction of the vector
  • Components along the x and y axes
  • Magnitude and angle with respect to a reference axis
Vector Components and Magnitude
\[\text{For vector } \vec{A} \text{ with magnitude } |\vec{A}| \text{ and angle } \theta:\] \[A_x = |\vec{A}| \cos \theta\] \[A_y = |\vec{A}| \sin \theta\] \[|\vec{A}| = \sqrt{A_x^2 + A_y^2}\] \[\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)\]
Where:
  • \(A_x\), \(A_y\) = x and y components of vector \(\vec{A}\)
  • \(|\vec{A}|\) = magnitude of vector \(\vec{A}\)
  • \(\theta\) = angle with positive x-axis

Position and Displacement

In two dimensions, the position of an object is described by its coordinates (x, y). Displacement is the change in position, which is a vector quantity that points from the initial to the final position.

Displacement in Two Dimensions
\[\vec{\Delta r} = (x_f - x_i, y_f - y_i) = (\Delta x, \Delta y)\] \[|\vec{\Delta r}| = \sqrt{(\Delta x)^2 + (\Delta y)^2}\] \[\theta = \tan^{-1}\left(\frac{\Delta y}{\Delta x}\right)\]
Where:
  • \(\vec{\Delta r}\) = displacement vector
  • \(x_f, y_f\) = final position coordinates
  • \(x_i, y_i\) = initial position coordinates
Example 1: Displacement Calculation

A hiker walks 3 km east and then 4 km north. Find the total displacement.

\(\Delta x = 3\) km, \(\Delta y = 4\) km

\[|\vec{\Delta r}| = \sqrt{(3 \text{ km})^2 + (4 \text{ km})^2} = \sqrt{9 + 16} = 5 \text{ km}\]

\[\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.1° \text{ north of east}\]

Velocity in Two Dimensions

Velocity in two dimensions is a vector that describes both the speed and direction of motion. It can be expressed in terms of its x and y components.

Velocity in Two Dimensions
\[\vec{v}_{avg} = \frac{\vec{\Delta r}}{\Delta t} = \left(\frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}\right) = (v_x, v_y)\] \[\vec{v} = \lim_{\Delta t \to 0} \frac{\vec{\Delta r}}{\Delta t} = \left(\frac{dx}{dt}, \frac{dy}{dt}\right)\] \[|\vec{v}| = \sqrt{v_x^2 + v_y^2}\] \[\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)\]
Where:
  • \(\vec{v}_{avg}\) = average velocity vector
  • \(\vec{v}\) = instantaneous velocity vector
  • \(|\vec{v}|\) = speed (magnitude of velocity)

Acceleration in Two Dimensions

Acceleration in two dimensions is the rate of change of velocity. Like velocity, it is a vector quantity with x and y components.

Acceleration in Two Dimensions
\[\vec{a}_{avg} = \frac{\vec{\Delta v}}{\Delta t} = \left(\frac{\Delta v_x}{\Delta t}, \frac{\Delta v_y}{\Delta t}\right) = (a_x, a_y)\] \[\vec{a} = \lim_{\Delta t \to 0} \frac{\vec{\Delta v}}{\Delta t} = \left(\frac{dv_x}{dt}, \frac{dv_y}{dt}\right)\] \[|\vec{a}| = \sqrt{a_x^2 + a_y^2}\] \[\theta = \tan^{-1}\left(\frac{a_y}{a_x}\right)\]

Projectile Motion

Projectile motion is a form of two-dimensional motion where an object is projected into the air and is subject only to the force of gravity (air resistance neglected). Examples include a thrown ball, a launched rocket, or water from a fountain.

Key features of projectile motion:

  • Horizontal motion occurs at constant velocity (\(a_x = 0\))
  • Vertical motion experiences constant downward acceleration due to gravity (\(a_y = -g\))
  • The horizontal and vertical motions are independent of each other
  • The path followed is a parabola
Projectile Motion Equations
\[\text{Horizontal Motion:}\] \[x = x_0 + v_{0x}t\] \[v_x = v_{0x} = \text{constant}\] \[\text{Vertical Motion:}\] \[y = y_0 + v_{0y}t - \frac{1}{2}gt^2\] \[v_y = v_{0y} - gt\]
Where:
  • \(v_{0x}\), \(v_{0y}\) = initial velocity components
  • \(g = 9.8\) m/s² = acceleration due to gravity
  • \(t\) = time

For an object projected from ground level at angle \(\theta\) with initial speed \(v_0\):

Projectile Motion Formulas
\[v_{0x} = v_0 \cos \theta, \quad v_{0y} = v_0 \sin \theta\] \[\text{Time of Flight: } T = \frac{2v_{0y}}{g} = \frac{2v_0 \sin \theta}{g}\] \[\text{Maximum Height: } h_{max} = \frac{v_{0y}^2}{2g} = \frac{v_0^2 \sin^2 \theta}{2g}\] \[\text{Horizontal Range: } R = v_{0x} \cdot T = \frac{v_0^2 \sin(2\theta)}{g}\]
Example 2: Projectile Motion

A ball is thrown with an initial velocity of 20 m/s at an angle of 30° above the horizontal. Calculate the maximum height and range.

Given: \(v_0 = 20\) m/s, \(\theta = 30°\)

\[v_{0x} = 20 \cos(30°) = 20 \times 0.866 = 17.32 \text{ m/s}\]

\[v_{0y} = 20 \sin(30°) = 20 \times 0.5 = 10 \text{ m/s}\]

\[h_{max} = \frac{v_{0y}^2}{2g} = \frac{(10)^2}{2(9.8)} = \frac{100}{19.6} \approx 5.1 \text{ m}\]

\[R = \frac{v_0^2 \sin(2\theta)}{g} = \frac{(20)^2 \sin(60°)}{9.8} = \frac{400 \times 0.866}{9.8} \approx 35.3 \text{ m}\]

Relative Motion

Relative motion describes how the motion of an object appears to an observer who may also be in motion. The relative velocity of object A with respect to object B is the vector difference of their velocities.

Relative Velocity
\[\vec{v}_{AB} = \vec{v}_A - \vec{v}_B\]
Where:
  • \(\vec{v}_{AB}\) = velocity of object A relative to object B
  • \(\vec{v}_A\) = velocity of object A in the reference frame
  • \(\vec{v}_B\) = velocity of object B in the reference frame
Example 3: Relative Motion

A boat moves at 5 m/s due east while a river flows at 3 m/s due south. Find the boat's velocity relative to the shore.

\(\vec{v}_{boat} = (5, 0)\) m/s, \(\vec{v}_{river} = (0, -3)\) m/s

\[\vec{v}_{boat,shore} = \vec{v}_{boat} + \vec{v}_{river} = (5, 0) + (0, -3) = (5, -3) \text{ m/s}\]

\[|\vec{v}_{boat,shore}| = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \text{ m/s}\]

\[\theta = \tan^{-1}\left(\frac{-3}{5}\right) \approx -31° \text{ (31° south of east)}\]

Applications of Two-Dimensional Motion

Understanding motion in two dimensions is essential for:

Sports Physics
  • Ball trajectories in baseball, golf, basketball
  • Optimizing throwing and kicking techniques
  • Understanding spin effects
Navigation
  • GPS systems and mapping
  • Aviation and maritime travel
  • Accounting for wind and current
Space & Military
  • Ballistics and trajectory calculations
  • Space mission planning
  • Satellite orbital mechanics
Entertainment
  • Video game physics engines
  • Animation and special effects
  • Virtual reality systems

Key Concepts Summary

  • Vectors have both magnitude and direction
  • Components: \(A_x = A \cos \theta\), \(A_y = A \sin \theta\)
  • Projectile motion: parabolic path with independent x and y motions
  • Horizontal motion: constant velocity (no acceleration)
  • Vertical motion: constant acceleration due to gravity
  • Relative motion: \(\vec{v}_{AB} = \vec{v}_A - \vec{v}_B\)
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