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3.1 Kinematics in Two Dimensions: An Introduction

2 min read•june 18, 2024

combines horizontal and vertical components, analyzed independently using equations. Gravity affects only the , while horizontal velocity remains constant without acceleration. This breakdown simplifies complex movements like .

The helps calculate total displacement and velocity in 2D motion. Perpendicular components are independent, allowing separate analysis. This principle applies to various scenarios, from thrown balls to objects on inclined planes.

Kinematics in Two Dimensions

Components of two-dimensional motion

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  • Two-dimensional motion broken down into horizontal (x) and vertical (y) components for analysis
    • Horizontal motion parallel to the ground (walking, running)
    • Vertical motion perpendicular to the ground (jumping, falling)
  • Each component analyzed independently using
    • Horizontal position: x=x0+v0xt+12axt2x = x_0 + v_{0x}t + \frac{1}{2}a_xt^2
    • Vertical position: y=y0+v0yt+12ayt2y = y_0 + v_{0y}t + \frac{1}{2}a_yt^2
  • (gg) only affects vertical component
    • Vertical acceleration: ay=−g=−9.8m/s2a_y = -g = -9.8 m/s^2 (negative indicates downward direction)
  • Horizontal velocity remains constant if no acceleration in x-direction (ax=0a_x = 0)
    • Constant horizontal velocity: vx=v0xv_x = v_{0x} (, ball thrown horizontally)

Pythagorean theorem for motion distances

  • Pythagorean theorem relates lengths of right triangle sides
    • Theorem: a2+b2=c2a^2 + b^2 = c^2, where cc is hypotenuse (longest side)
  • Calculate or velocity in two-dimensional motion
    • Resultant displacement: Δr=(Δx)2+(Δy)2\Delta r = \sqrt{(\Delta x)^2 + (\Delta y)^2} (total distance traveled)
    • : v=vx2+vy2v = \sqrt{v_x^2 + v_y^2} (total speed and direction)
  • Find angle of resultant vector using trigonometric functions
    • : tan⁡θ=ΔyΔx\tan \theta = \frac{\Delta y}{\Delta x} (direction of motion)
    • : tan⁡θ=vyvx\tan \theta = \frac{v_y}{v_x} (direction of velocity)
  • Vector addition is used to combine multiple displacements or velocities in two dimensions

Independence of perpendicular motions

  • Horizontal and vertical components of motion independent of each other
    • Changes in one component do not affect the other (ball thrown at an angle)
  • Independence due to perpendicular nature of components
    • Perpendicular have no influence on each other (x and y axes)
  • Examples of independent perpendicular motions:
    1. Projectile's horizontal velocity constant, vertical velocity changes due to gravity (cannon ball, basketball shot)
    2. Object sliding down frictionless ramp has constant acceleration in ramp direction, perpendicular component remains zero (skier on slope, block on incline)

Projectile Motion

  • describes the curved path of a projectile under the influence of gravity
  • is the path followed by a projectile through space
  • is the horizontal distance traveled by a projectile
  • is the total time a projectile remains in the air

Key Terms to Review (21)

Acceleration Due to Gravity: Acceleration due to gravity, often denoted as 'g', is the acceleration experienced by an object due to the Earth's gravitational pull. This constant acceleration acts on all objects near the Earth's surface, causing them to experience a downward force and a change in velocity over time.
Displacement Angle: The displacement angle is the angle between the initial position vector and the final position vector of an object moving in two-dimensional space. It represents the direction of the object's displacement from its starting point to its ending point.
Horizontal Component: The horizontal component of a vector or quantity is the projection of that vector or quantity onto the horizontal axis. It represents the portion of the vector or quantity that is parallel to the ground or a horizontal surface.
Independence of Perpendicular Motions: The principle that the motion of an object in one direction is independent of and does not affect its motion in a perpendicular direction. This concept is central to understanding the kinematics of two-dimensional motion.
Kinematics: Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion. It focuses on parameters such as position, velocity, acceleration, and time, allowing us to describe how an object moves in space over time and understand various forms of motion.
One-Dimensional Kinematics Equations: One-dimensional kinematics equations are a set of fundamental equations used to describe the motion of an object along a straight line. These equations relate the position, velocity, acceleration, and time of an object's motion in a single dimension.
Parabolic Motion: Parabolic motion refers to the curved trajectory followed by an object that is projected or launched into the air, such as a ball, projectile, or other moving body, under the influence of gravity and without significant air resistance. This type of motion is characterized by a combination of horizontal and vertical components, resulting in a parabolic path.
Projectile motion: Projectile motion describes the trajectory of an object that is subject only to the acceleration of gravity. It involves both horizontal and vertical components of motion.
Projectile Motion: Projectile motion is the motion of an object that is launched into the air and follows a curved trajectory under the influence of gravity. It is a type of motion that involves both horizontal and vertical components, and is governed by the laws of kinematics and Newton's laws of motion.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Range: Range is the horizontal distance traveled by a projectile from its launch point to the landing point. It depends on the initial velocity and angle of projection.
Range: The range of a quantity is the difference between its maximum and minimum values. It represents the extent or spread of a set of data or measurements and is a measure of the dispersion or variability within the data.
Resultant Displacement: Resultant displacement is the net or combined displacement experienced by an object that is undergoing motion in two or more dimensions. It represents the overall change in position of the object from its initial location to its final location, taking into account the individual displacements along each axis.
Resultant Velocity: Resultant velocity is the overall or net velocity of an object when two or more velocities are combined. It represents the final velocity that an object experiences due to the vector addition of individual velocity components.
Time of Flight: Time of flight refers to the duration of time it takes for an object, such as a projectile or a particle, to travel from its initial position to its final position. This concept is particularly important in the study of kinematics, which is the branch of physics that deals with the motion of objects without considering the forces that cause the motion.
Trajectory: Trajectory is the path that a moving object follows through space as a function of time. It is determined by factors such as initial velocity, angle of launch, and the forces acting on the object (e.g., gravity).
Trajectory: A trajectory is the path or curve that an object follows through space over time. It describes the motion and position of an object as it moves from one point to another, taking into account factors such as initial position, velocity, acceleration, and the effects of forces acting on the object.
Two-Dimensional Motion: Two-dimensional motion refers to the movement of an object in a plane, where the object's position and trajectory can be described using two spatial dimensions, typically the horizontal (x) and vertical (y) axes. This type of motion is an essential concept in the study of kinematics, which is the branch of physics that deals with the motion of objects without considering the forces that cause the motion.
Vectors: Vectors are quantities that have both magnitude and direction. They are used to represent physical quantities such as displacement, velocity, and electric field.
Velocity Angle: The velocity angle is the angle between an object's velocity vector and a reference direction, typically the positive x-axis. It describes the direction of an object's motion in two-dimensional space.
Vertical Component: The vertical component of a vector or quantity refers to the portion or magnitude of that vector or quantity that is oriented in the vertical or up-down direction. It represents the y-coordinate or height-related aspect of a vector or motion, independent of the horizontal or left-right direction.
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3.2 Vector Addition and Subtraction: Graphical Methods