Glossary

2.2 Velocity and Acceleration

2 min read•july 24, 2024

and are key concepts in understanding motion. They describe how objects move and change over time, helping us analyze everything from car trips to falling objects.

Graphs bring these concepts to life, showing patterns in motion visually. By interpreting slopes and areas under curves, we can extract valuable information about an object's behavior, making complex motion easier to understand and predict.

Kinematics in One Dimension

Average vs instantaneous velocity

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  • Velocity measures rate of change of position with respect to time as vector quantity (magnitude and direction)
  • vavg=ΔxΔtv_{avg} = \frac{\Delta x}{\Delta t} represents overall motion between two points using and time interval (car trip)
  • v=lim⁡Δt→0ΔxΔt=dxdtv = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} describes velocity at specific moment calculated as limit of average velocity as time interval approaches zero (speedometer reading)
  • Both measured in m/s

Concept of acceleration

  • Acceleration measures rate of change of velocity with respect to time as vector quantity (magnitude and direction)
  • aavg=ΔvΔta_{avg} = \frac{\Delta v}{\Delta t} describes overall change in velocity over time interval
  • a=lim⁡Δt→0ΔvΔt=dvdta = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} gives acceleration at specific moment
  • Positive acceleration increases velocity (car speeding up), negative acceleration decreases velocity (car braking)
  • Affects rate of change of position as second derivative a=d2xdt2a = \frac{d^2x}{dt^2}
  • Measured in m/s²

Velocity and acceleration calculations

  • for one-dimensional motion:
    1. v=v0+atv = v_0 + at
    2. x=x0+v0t+12at2x = x_0 + v_0t + \frac{1}{2}at^2
    3. v2=v02+2a(x−x0)v^2 = v_0^2 + 2a(x - x_0)
    4. x=x0+12(v+v0)tx = x_0 + \frac{1}{2}(v + v_0)t
  • Problem-solving steps:
    1. Identify known and unknown variables
    2. Choose appropriate kinematic equation
    3. Solve for unknown variable
    4. Check units and reasonableness of answer
  • Special cases include constant velocity motion (a = 0) and (a = g = 9.8 m/s²)

Graphs of motion in one dimension

  • Velocity-time graphs:
    • Slope represents acceleration
    • Area under curve gives displacement
    • Horizontal line shows constant velocity (cruise control)
    • Positive slope indicates increasing velocity (car accelerating)
    • Negative slope shows decreasing velocity (car slowing down)
  • Acceleration-time graphs:
    • Area under curve gives change in velocity
    • Horizontal line indicates constant acceleration (free fall)
    • Positive slope shows increasing acceleration (rocket launch)
    • Negative slope indicates decreasing acceleration (parachute opening)
  • is integral of
  • Acceleration-time graph is derivative of velocity-time graph
  • Create graphs by plotting data points, connecting with appropriate curve or line, labeling axes with units
  • Interpret graphs by identifying periods of constant, increasing, or decreasing velocity/acceleration, determining initial and final values, calculating average values over specific time intervals

Key Terms to Review (18)

A = δv/δt: The equation $$a = \frac{\delta v}{\delta t}$$ defines acceleration as the rate of change of velocity over time. This fundamental concept is crucial in understanding how an object's speed and direction change, linking motion dynamics to the forces that cause such changes. It emphasizes that acceleration can occur due to variations in either speed or direction, or both, which are vital in analyzing any object's motion.
Acceleration: Acceleration is the rate at which an object's velocity changes over time. It describes how quickly an object speeds up, slows down, or changes direction, and it is directly related to the forces acting on that object as described by fundamental laws of motion.
Acceleration-time graph: An acceleration-time graph is a visual representation that displays how the acceleration of an object changes over time. The graph typically plots acceleration on the vertical axis and time on the horizontal axis, allowing for a clear view of an object's acceleration behavior during its motion. It helps in understanding how forces affect an object's motion and provides insights into various phases of movement, such as speeding up or slowing down.
Average acceleration: Average acceleration is defined as the change in velocity of an object over a specified time interval. It quantifies how quickly an object's velocity changes, highlighting the relationship between initial and final velocities in that time frame. This concept is crucial in understanding motion and is essential for calculating how fast an object speeds up or slows down during its journey.
Average velocity: Average velocity is defined as the total displacement divided by the total time taken for that displacement. It reflects the overall change in position of an object and is a vector quantity, which means it has both magnitude and direction. Understanding average velocity is essential when analyzing motion, as it helps to differentiate between how fast something moves versus the direction in which it moves.
Displacement: Displacement is defined as the shortest distance from an object's initial position to its final position, along with the direction of that straight line. It’s a vector quantity, meaning it has both magnitude and direction, which distinguishes it from scalar quantities that only measure magnitude. Displacement helps in understanding how far and in what direction an object has moved, playing a vital role in analyzing motion, velocity, and energy in oscillatory systems.
Free Fall: Free fall refers to the motion of an object under the influence of gravitational force only, without any other forces acting on it, like air resistance. During free fall, objects accelerate towards the Earth at a constant rate due to gravity, which connects to how motion is described and analyzed, as well as the energy transformations involved in such movements.
Instantaneous acceleration: Instantaneous acceleration refers to the rate of change of velocity of an object at a specific moment in time. This concept is crucial in understanding motion, as it allows us to describe how quickly an object's speed or direction is changing at any given instant. Instantaneous acceleration can vary from one moment to the next, depending on the forces acting on the object, making it a fundamental aspect of kinematics and dynamics.
Instantaneous velocity: Instantaneous velocity is defined as the rate of change of an object's position at a specific moment in time. It indicates how fast an object is moving and in which direction at that particular instant, providing a more precise measure than average velocity over a time interval. Instantaneous velocity is represented mathematically as the derivative of the position function with respect to time, highlighting its connection to both displacement and time.
Kinematic Equations: Kinematic equations are a set of four fundamental equations that describe the motion of an object under uniform acceleration. They relate the object's displacement, initial and final velocity, acceleration, and time, allowing for the prediction of future motion based on initial conditions. These equations are essential for understanding various types of motion, including straight-line motion, free fall, and projectile motion.
Meters per second (m/s): Meters per second (m/s) is the SI unit of measurement for velocity, indicating the distance in meters that an object travels in one second. This unit allows for a clear understanding of how fast an object is moving and is essential for analyzing motion in various contexts. It's commonly used in equations to describe both speed and velocity, which take direction into account when needed.
Meters per second squared (m/s²): Meters per second squared (m/s²) is the unit of measurement for acceleration in the International System of Units (SI). It describes how quickly an object's velocity changes over time, indicating both the speed increase or decrease and the direction of that change. Understanding this unit is essential for analyzing motion, particularly when it comes to concepts like velocity and acceleration, as it quantifies the rate at which an object accelerates or decelerates.
Non-uniform acceleration: Non-uniform acceleration refers to a situation where an object's acceleration changes over time, meaning the rate of change of velocity is not constant. This can occur in various forms of motion, such as when an object speeds up or slows down in a non-linear way, resulting in varying accelerations at different points in time. Understanding non-uniform acceleration is crucial as it relates to the analysis of motion, allowing for more accurate predictions and calculations in physics.
Speed: Speed is a scalar quantity that represents how fast an object moves, calculated as the distance traveled divided by the time taken to travel that distance. It provides a basic measure of motion, without considering the direction, and is typically expressed in units such as meters per second (m/s) or kilometers per hour (km/h). Understanding speed is crucial for analyzing movement in various contexts, including changes in velocity and acceleration.
Uniform acceleration: Uniform acceleration is the condition in which an object's velocity changes at a constant rate over time. This means that the object's speed increases or decreases evenly, leading to a straight-line motion when graphed. In this context, it’s essential to understand how it relates to concepts of velocity and acceleration, as well as the behavior of freely falling objects, where the acceleration due to gravity remains constant.
V = d/t: The equation v = d/t represents the relationship between velocity (v), distance (d), and time (t). In this formula, velocity is defined as the rate at which an object changes its position over a certain period of time. This relationship is fundamental in understanding how quickly an object is moving and lays the groundwork for analyzing motion and acceleration in various contexts.
Velocity: Velocity is a vector quantity that describes the rate at which an object changes its position. It includes both the speed of the object and the direction in which it moves, making it distinct from speed, which is a scalar quantity. Understanding velocity is crucial when analyzing how objects move in various contexts, such as linear motion or more complex scenarios like oscillations.
Velocity-time graph: A velocity-time graph is a visual representation of an object's velocity over time, where the x-axis represents time and the y-axis represents velocity. This type of graph helps in analyzing motion by providing clear insights into an object's speed and direction, allowing us to see how velocity changes throughout the motion. The slope of the graph indicates acceleration, while the area under the curve can represent displacement.
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