Learning Outcomes
i. Understand the concept of instantaneous velocity and its significance in motion analysis.
ii. Differentiate between instantaneous velocity and average velocity.
iii. Identify the tangent line to a displacement-time graph at a specific point.
iv. Calculate the slope of the tangent line as the instantaneous velocity at that point.
v. Interpret the instantaneous velocity obtained from the slope of the tangent line.
Introduction
In the realm of physics, understanding the motion of objects requires a precise measure of their velocity at a specific instant, not just an average over a time interval. This instantaneous velocity provides a snapshot of how fast and in which direction an object is moving at a particular point in time.
i. Instantaneous Velocity: Capturing Motion at an Instant
Instantaneous velocity, denoted by v, represents the velocity of an object at a specific point in time. It is the limit of average velocity as the time interval approaches zero. In other words, instantaneous velocity provides a precise measure of the object's speed and direction at an instant.
ii. Distinguishing between Instantaneous and Average Velocity
Average velocity, denoted by v̄, represents the total displacement of an object divided by the time interval during which the displacement occurred. It provides a general idea of the object's motion over a period, but it does not capture the exact velocity at a specific point.
iii. Measuring Instantaneous Velocity from Displacement-Time Graph
A displacement-time graph provides a visual representation of an object's motion. The slope of the graph at any point represents the instantaneous velocity at that point. The steeper the slope, the faster the object is moving at that instant.
Identifying the Tangent Line
To determine the instantaneous velocity at a specific point on the displacement-time graph, we construct a tangent line to the graph at that point. A tangent line is a line that touches the curve at a single point and has the same slope as the curve at that point.
Calculating the Slope of the Tangent Line
The slope of the tangent line represents the instantaneous velocity at the point of contact. It is calculated as the change in displacement (Δr) divided by the change in time (Δt):
Slope = Δr/Δt = Instantaneous Velocity (v)
A positive slope indicates motion in the positive direction, while a negative slope indicates motion in the negative direction. The magnitude of the slope represents the speed of the object at that instant.
iv. Interpreting Instantaneous Velocity
The instantaneous velocity obtained from the slope of the tangent line provides valuable insights into the object's motion at that specific point in time. It allows us to determine the exact speed and direction of the object's movement at that instant.
Determining instantaneous velocity from a displacement-time graph is an essential technique in analyzing motion. By measuring the slope of the tangent line at a specific point, we gain a precise understanding of how fast and in which direction an object is moving at that instant. This concept is crucial in various physical scenarios, including projectile motion, uniform motion, and accelerated motion.
