Principles of Physics I Unit 2 Review: Kinematics in One Dimension

Key Concepts and Definitions

• Kinematics studies the motion of objects without considering the forces causing the motion
• Scalar quantities have magnitude only and no direction (speed, distance, time)
• Vector quantities have both magnitude and direction (velocity, acceleration, displacement)
• Reference point serves as the origin (starting point) for position measurements
• Coordinate system consists of a fixed reference point (origin) and a set of axes (x, y, z) used to specify positions
  • One-dimensional motion occurs along a straight line and requires only one coordinate axis
  • Two-dimensional motion occurs in a plane and requires two coordinate axes
  • Three-dimensional motion occurs in space and requires three coordinate axes

Motion in a Straight Line

• Motion along a straight line path is one-dimensional motion
• Position is the location of an object relative to a chosen reference point (origin)
• Position can be positive or negative depending on the direction from the origin
• Displacement measures the change in position of an object
  • Displacement = Final Position - Initial Position
  • Denoted by the symbol $\Delta x$ where $\Delta$ represents change
• Distance is the total length of the path traveled by an object regardless of direction
• Speed measures how fast an object moves and is the distance traveled per unit time
• Velocity measures the rate and direction of change in position (speed with direction)

Position, Displacement, and Distance

• Position is a vector quantity that specifies an object's location relative to a reference point
• Position can be represented using a coordinate system (x-axis for one-dimensional motion)
• Displacement measures the shortest distance between the initial and final positions and includes direction
  • Displacement = $\Delta x = x_f - x_i$, where $x_f$ is the final position and $x_i$ is the initial position
• Distance is a scalar quantity that measures the total path length traveled regardless of direction
• Displacement and distance are equal only when an object moves in a straight line in one direction
• Displacement can be positive, negative, or zero, while distance is always positive

Velocity and Speed

• Velocity is a vector quantity that measures the rate and direction of change in position
  • Average velocity = $v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$
• Speed is a scalar quantity that measures the rate of motion without considering direction
  • Average speed = $s_{avg} = \frac{Total\,distance}{Total\,time}$
• Instantaneous velocity is the velocity at a specific instant in time
  • Instantaneous velocity = $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$
• Velocity can be positive (motion in the positive direction), negative (motion in the negative direction), or zero (object at rest)
• Speed is always positive or zero and does not include direction

Acceleration

• Acceleration is a vector quantity that measures the rate of change of velocity
  • Average acceleration = $a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$
• Acceleration can be positive (velocity increasing), negative (velocity decreasing), or zero (constant velocity)
• Instantaneous acceleration is the acceleration at a specific instant in time
  • Instantaneous acceleration = $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$
• Acceleration due to gravity ($g$) is the acceleration experienced by objects in free fall near Earth's surface
  • $g \approx 9.8\,m/s^2$ (downward)
• Deceleration is the term used when an object slows down (negative acceleration)

Equations of Motion

• Equations of motion relate position, velocity, acceleration, and time for constant acceleration
• $v = v_0 + at$
  • $v$ is the final velocity, $v_0$ is the initial velocity, $a$ is the constant acceleration, and $t$ is the time
• $x = x_0 + v_0t + \frac{1}{2}at^2$
  • $x$ is the final position, $x_0$ is the initial position
• $v^2 = v_0^2 + 2a(x - x_0)$
  • Relates final velocity, initial velocity, acceleration, and displacement without time
• These equations are valid only for constant acceleration and one-dimensional motion
• When using equations of motion, choose a coordinate system and consistently use positive or negative signs for direction

Graphs in Kinematics

• Graphs help visualize the relationships between position, velocity, and acceleration over time
• Position-time graphs show an object's position as a function of time
  • Slope of the tangent line at any point represents the instantaneous velocity
  • Area under the curve represents the displacement between two times
• Velocity-time graphs show an object's velocity as a function of time
  • Slope of the tangent line at any point represents the instantaneous acceleration
  • Area under the curve represents the displacement between two times
• Acceleration-time graphs show an object's acceleration as a function of time
  • Area under the curve represents the change in velocity between two times

Problem-Solving Strategies

• Read the problem carefully and identify the given information and the quantity to be found
• Draw a diagram or sketch of the situation, establishing a coordinate system and positive direction
• List the known quantities and assign variables to the unknown quantities
• Determine which equation(s) of motion are appropriate based on the given information and the unknown quantity
• Solve the equation(s) for the unknown quantity, substituting the known values and simplifying
• Check the units of the final answer to ensure they are consistent with the quantity being solved for
• Analyze the result to see if it makes sense in the context of the problem (reasonable magnitude and sign)