College Physics II – Mechanics, Sound, Oscillations, and Waves
Definition
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.
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The instantaneous velocity v(t) is the derivative of the object's position function with respect to time.
v(t) represents the slope of the tangent line to the object's position-time graph at a specific point.
The units of v(t) are typically meters per second (m/s) in the International System of Units (SI).
Instantaneous velocity can be positive, negative, or zero, depending on the direction of the object's motion.
Changes in v(t) over time indicate the presence of acceleration, which is the derivative of velocity with respect to time.
Review Questions
Explain how the concept of v(t) relates to the idea of instantaneous velocity.
The term v(t) represents the instantaneous velocity of an object, which is the rate of change of its position at a specific moment in time. Instantaneous velocity is calculated as the limit of the average velocity as the time interval approaches zero, and v(t) provides a mathematical expression of this instantaneous rate of change. By understanding v(t), you can determine the object's speed and direction of motion at any given point during its movement.
Describe the relationship between v(t) and the object's position-time graph.
The instantaneous velocity v(t) is directly related to the slope of the tangent line to the object's position-time graph at a specific point. The steeper the slope of the tangent line, the greater the value of v(t) at that instant. Conversely, if the tangent line is horizontal (i.e., the slope is zero), then v(t) is also zero, indicating the object is momentarily at rest. Understanding this connection between v(t) and the position-time graph is crucial for analyzing and interpreting the motion of an object.
Analyze how changes in v(t) over time can provide information about the object's acceleration.
The rate of change of an object's velocity, known as acceleration, is the derivative of the instantaneous velocity v(t) with respect to time. If v(t) is increasing over time, the object is experiencing positive acceleration, meaning its velocity is increasing. If v(t) is decreasing over time, the object is experiencing negative acceleration, or deceleration, as its velocity is decreasing. Analyzing the changes in v(t) can therefore reveal important information about the object's acceleration and the forces acting upon it during its motion.