3.2 Instantaneous Velocity and Speed
3 min read•june 24, 2024
Velocity can be tricky to grasp, but it's crucial for understanding motion. We'll explore the difference between average and , and how they relate to an object's speed and direction.
helps us pinpoint an object's exact velocity at any moment. We'll see how derivatives let us calculate from position equations, and apply this to real-world scenarios like car crashes and roller coasters.
Instantaneous Velocity and Speed
Average vs instantaneous velocity
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- calculated as change in position divided by change in time () represents overall velocity between two points
- Does not provide information about velocity at a specific instant
- Instantaneous velocity is velocity of an object at a particular moment in time
- Represents of position with respect to time at a specific point
- Determined by finding of as time interval approaches zero ()
- Instantaneous velocity is derivative of position with respect to time ()
Velocity and speed relationship
- Both velocity and speed measure rate at which an object moves expressed in units of distance per unit time ()
- Velocity is a with both magnitude and direction while speed is a with only magnitude
- Magnitude of velocity is equal to speed of an object
- Object's speed is always non-negative while velocity can be positive, negative, or zero
- Negative velocity indicates motion in opposite direction of chosen positive axis (moving backwards)
- Zero velocity means object is not moving (stationary)
Calculating instantaneous velocity
- Given position equation , instantaneous velocity found by taking derivative with respect to time
- Example: If position of an object is , instantaneous velocity is
- To find instantaneous velocity at a specific time, substitute time value into velocity equation
- For example above, at seconds, instantaneous velocity would be m/s
- Instantaneous velocity can be visualized as the of the to the at a specific point
Applications of instantaneous velocity
- Instantaneous velocity used to analyze motion in various scenarios such as determining speed of a vehicle at a specific moment, calculating velocity of a projectile at a particular point in its trajectory (bullet fired from a gun), or analyzing motion of objects in free fall or under influence of forces (skydiver)
- When solving problems, important to identify given information (position equation or relevant data points), determine appropriate equations or methods to use based on problem statement, perform calculations accurately, and interpret results in context of the problem
- Real-world examples:
- Calculating speed of a car at the moment of a collision for accident reconstruction
- Determining velocity of a spacecraft at a specific point in its orbit around Earth
- Analyzing velocity of a roller coaster at the top of a loop to ensure rider safety
Calculus and instantaneous velocity
- Calculus provides tools to analyze in various physical phenomena
- Limit concept in calculus allows for precise definition of instantaneous velocity
- Derivative represents instantaneous rate of change, which is fundamental to understanding instantaneous velocity
- Slope of the tangent line to a position-time graph gives the instantaneous velocity at that point
Key Terms to Review (31)
Acceleration: Acceleration is the rate of change of velocity with respect to time. It represents the change in an object's speed or direction over a given time interval, and is a vector quantity that has both magnitude and direction.
Average velocity: Average velocity is the total displacement divided by the total time taken. It is a vector quantity with both magnitude and direction.
Average Velocity: Average velocity is a measure of the average rate of change in an object's position over a given time interval. It represents the total displacement of an object divided by the total time elapsed, providing a single value that summarizes the object's motion during that time period.
Calculus: Calculus is a branch of mathematics that deals with the study of rates of change and the accumulation of quantities. It is a powerful tool for analyzing and understanding the behavior of dynamic systems, such as the motion of objects and the growth of populations.
Differentiation: Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function at a specific point. It is a fundamental concept in calculus that is used to analyze the behavior of functions and solve a wide range of problems in physics, engineering, and other scientific fields.
Displacement: Displacement is a vector quantity that refers to the change in position of an object. It is measured as the straight-line distance from the initial to the final position, along with the direction.
Displacement: Displacement is the change in position of an object relative to a reference point. It is a vector quantity, meaning it has both magnitude and direction, and is used to describe the movement of an object in physics.
Dx/dt: The derivative of a function with respect to the independent variable, typically denoted as 'dx/dt', represents the instantaneous rate of change of the function at a specific point. It is a fundamental concept in calculus that is essential for understanding the topics of instantaneous velocity and speed.
Free-Fall: Free-fall is the motion of an object under the sole influence of gravity, where the object is falling without any other forces acting upon it. This motion is characterized by a constant acceleration due to the Earth's gravitational pull, resulting in a predictable and uniform rate of change in the object's velocity over time.
Instantaneous Rate of Change: The instantaneous rate of change refers to the rate of change of a quantity at a specific point in time, capturing the immediate or instantaneous change rather than the average change over a finite interval. It is a fundamental concept in calculus that describes the slope or derivative of a function at a particular point.
Instantaneous speed: Instantaneous speed is the magnitude of the instantaneous velocity. It represents how fast an object is moving at a specific moment in time.
Instantaneous velocity: Instantaneous velocity is the velocity of an object at a specific moment in time. It is the derivative of the object's position with respect to time.
Instantaneous Velocity: Instantaneous velocity is the rate of change of an object's position at a specific moment in time. It represents the object's speed and direction of motion at an infinitesimally small interval, providing a precise measure of the object's motion at that instant.
Limit: The limit of a function or sequence is the value that the function or sequence approaches as the input variable approaches a certain value or as the sequence progresses. It represents the behavior of a function or sequence at a specific point or as it approaches infinity.
Meters per Second: Meters per second (m/s) is a unit of measurement that represents the rate of change in position over time. It is commonly used to express the velocity or speed of an object, indicating the distance traveled per unit of time.
Position-Time Graph: A position-time graph is a graphical representation that shows the position of an object as a function of time. It is a fundamental tool used to analyze and understand the motion of an object in physics, particularly in the context of kinematics, which is the study of motion without considering the forces that cause it.
Projectile motion: Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration due to gravity. It involves two components of motion: horizontal and vertical.
Projectile Motion: Projectile motion is the motion of an object that is launched into the air and moves solely under the influence of gravity and without any additional force acting on it. It is a type of motion that follows a curved trajectory, with the object's position and velocity changing over time in a predictable manner.
Rate of Change: The rate of change is a measure of how quickly a quantity is changing over time. It describes the change in a variable divided by the change in another variable, typically time, and represents the slope or steepness of a line or curve.
Scalar Quantity: A scalar quantity is a physical quantity that is fully described by a single numerical value and a unit. It has magnitude, or size, but no direction associated with it. Scalar quantities are often contrasted with vector quantities, which have both magnitude and direction.
Slope: Slope is a measure of the steepness or incline of a line or curve, representing the rate of change in the vertical direction (y-coordinate) with respect to the horizontal direction (x-coordinate). It is a fundamental concept in various fields, including physics, mathematics, and engineering.
Tangent Line: A tangent line is a straight line that touches a curve at a single point, without crossing or intersecting the curve at that point. It represents the instantaneous rate of change of the function at that specific point.
Uniform Motion: Uniform motion is a type of motion where an object travels at a constant speed, maintaining the same velocity throughout its journey. This means the object covers equal distances in equal intervals of time, with no acceleration or deceleration involved.
V_{avg}: The average velocity, denoted as $v_{avg}$, is a measure of the average rate of change of an object's position over a given time interval. It represents the total displacement of an object divided by the time taken to cover that displacement.
V_{inst}: The symbol v_{inst} represents instantaneous velocity, which is the velocity of an object at a specific moment in time. This concept helps in understanding how fast an object is moving and in which direction at that exact instant, distinguishing it from average velocity which considers the overall displacement over a time interval. Instantaneous velocity is crucial for analyzing motion because it provides a more precise measurement of an object's motion at any given moment.
V(t): v(t) represents the instantaneous velocity of an object as a function of time. It describes the rate of change of an object's position over time, providing a measure of how quickly the object is moving at a specific moment.
Vector Quantity: A vector quantity is a physical measurement that has both magnitude and direction, distinguishing it from scalar quantities that have only magnitude. Vector quantities are essential in physics as they provide a complete description of various physical phenomena, such as motion and forces. Understanding vector quantities allows for better analysis of how objects move and interact in space.
Velocity-Time Graph: A velocity-time graph is a graphical representation that depicts the relationship between an object's velocity and time. It is a fundamental tool in understanding and analyzing the motion of an object, as it provides a visual representation of the object's speed and direction of motion over a given time period.
X(t): The notation x(t) represents the position of an object as a function of time, indicating how the object's location changes over time. It provides a mathematical framework to analyze motion, where 'x' denotes the position along a specified axis and 't' represents time. Understanding x(t) is essential for calculating instantaneous velocity and speed, as it allows one to determine how quickly an object moves and in what direction at any given moment.
Δt: Δt, or delta t, represents the change in time between two different instances or events. It is a fundamental concept in the study of motion and acceleration, as it quantifies the time interval over which changes in position, velocity, and acceleration occur.
Δx: Δx, or delta x, represents the change in position or displacement of an object over a given time interval. It is a fundamental concept in the study of kinematics, the branch of physics that describes the motion of objects without considering the forces that cause the motion.