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plot-position-time-graph-for-non-uniform-motion

🚀 Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. When analyzing motion in a straight line, we often use position-time graphs to represent how the position of an object changes over time. In the case of non-uniform motion, the velocity of the object changes, which results in a curve on the position-time graph. This graph allows us to visualize the object's position at various times and to interpret the motion characteristics, such as acceleration and deceleration.

Theory Explanation

Understanding Position-Time Graphs

A position-time graph plots the position of an object on the y-axis against time on the x-axis. For non-uniform motion, the graph will not be a straight line but rather a curve. The slope of the graph at any point represents the instantaneous velocity of the object. If the graph is curving upwards, the object is accelerating; if it is curving downwards, the object is decelerating.

Calculating Average Velocity

The average velocity can be calculated using the formula: \( v_{avg} = \frac{\Delta x}{\Delta t} \), where \( \Delta x \) is the change in position and \( \Delta t \) is the change in time. This can be derived from the coordinates of two points on the position-time graph.

\[ v_{avg} = \frac{\Delta x}{\Delta t} \]
Analyzing the Graph

To analyze a position-time graph for non-uniform motion, look for changes in the curvature. A steeper slope indicates a higher speed, while a flatter slope indicates a lower speed. The shape of the curve can provide insights into the acceleration of the object. If the curve is getting steeper, the object is accelerating; if it is flattening, the object is decelerating.

Key Points

  • 🎯 Position-time graphs represent the position of an object over time.
  • 🎯 Non-uniform motion results in a curved position-time graph.
  • 🎯 The slope of the graph indicates the velocity of the object at a given time.

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Examples:💡

An object moves in a straight line, and its position at different times is given as follows: At t=0s, x=0m; at t=2s, x=4m; at t=4s, x=10m; at t=6s, x=18m. Plot the position-time graph and analyze the motion.

Solution:

Step 1: Plot the points on a graph: (0,0), (2,4), (4,10), (6,18).

Step 2: Connect the points to visualize the curve. Notice how the slope changes, indicating non-uniform motion.

Step 3: Calculate the average velocity between t=0s and t=6s: \( v_{avg} = \frac{18m - 0m}{6s - 0s} = 3m/s \).

\[ v_{avg} = \frac{18m}{6s} = 3m/s \]

A car accelerates from rest and its position at t=0s is 0m, at t=3s is 9m, at t=6s is 24m. Plot the position-time graph and find the average velocity.

Solution:

Step 1: Plot the points: (0,0), (3,9), (6,24).

Step 2: Connect the points to show the curve of acceleration.

Step 3: Calculate average velocity from t=0s to t=6s: \( v_{avg} = \frac{24m - 0m}{6s - 0s} = 4m/s \).

\[ v_{avg} = \frac{24m}{6s} = 4m/s \]

Common Mistakes

  • Mistake: Students may confuse the slope of the graph with average speed instead of instantaneous velocity.

    Correction: Emphasize that the slope at any point on the curve represents instantaneous velocity, while the overall slope between two points gives average speed.

  • Mistake: Failing to accurately plot points can lead to misinterpretation of the graph.

    Correction: Encourage careful plotting of points and checking coordinates before connecting them.

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