ANAND CLASSES study material and notes to learn about the Position-Time Graph in kinematics, its significance, types, and interpretations. Explore conceptual questions, MCQs, worksheets, and a test paper to master this essential topic for JEE, NEET, and CBSE Class 11 exams.
INTRODUCTION
In kinematics, the motion of a particle is analyzed through various parameters such as initial velocity u, final velocity v, acceleration a, and displacement s. The Position-Time (x-t) graph is a fundamental tool for understanding motion, where time tt is plotted on the x-axis and position x on the y-axis.
A position-time graph helps in visualizing how an object moves over time. The slope of this graph provides crucial information about the velocity of the object. By analyzing the graph, we can determine whether the object is at rest, moving with constant velocity, or accelerating.
Understanding Position-Time Graph
Slope of position-time graph represents the velocity of the particle
The Position-Time Graph (x-t graph) shows how the position of a particle varies with time.
The slope of this graph represents the velocity of the particle.
The equation for velocity is given by: $$v = \frac{\text{Change in position}}{\text{Time taken}} = \frac{y_2 – y_1}{t_2 – t_1}$$
Using trigonometry, in a right-angled triangle: $$\tan \theta = \frac{BC}{AC} = \frac{AD}{AC} = \frac{y_2 – y_1}{t_2 – t_1}$$
Thus, velocity is given by: $$v = \tan \theta$$
It is clear that slope of position-time graph represents the velocity of the particle.
A steeper slope means higher velocity, while a flat slope (zero slope) represents zero velocity.
Various Types of Position-Time Graphs and Their Interpretation
1. Straight Line with Positive Slope (Uniform Motion)
Represents uniform motion with constant velocity.
The object moves equal distances in equal intervals of time.
The equation of motion in this case is: It is clear that slope of position-time graph represents the velocity of the particle. $$x = x_0 + vt$$
Example: A car moving at a constant speed.
2. Straight Line with Zero Slope (Object at Rest)
θ = 0o so $v$ = 0 i.e., line parallel to time axis represents that the particle is at rest.
Represents a stationary object.
The position remains unchanged over time.
Equation: $$x = x_0 \:(constant \: position)$$
Example: A parked car.
3. Curved Graph (Non-Uniform Motion: Acceleration or Deceleration)
Represents accelerated or decelerated motion.
A concave-up curve indicates positive acceleration(θ is increasing so $v$ is increasing, $a$ is positive.).
A concave-down curve indicates negative acceleration (deceleration) i.e.(θ is decreasing so $v$ is decreasing, $a$ is negative).
Example: A ball rolling down a slope.
Exam-Oriented Question-Answer Format
Q1: What does the slope of a position-time graph represent?
A1: The slope of a position-time graph represents the velocity of the particle. A steeper slope indicates a higher velocity, while a zero slope means the object is stationary.
Q2: How can we determine if an object is at rest using a position-time graph?
A2: If the graph is a horizontal straight line, the object is at rest as its position does not change with time.
Q3: What does a curved position-time graph indicate?
A3: A curved position-time graph indicates non-uniform motion, meaning the object is accelerating or decelerating.
Multiple Choice Questions (MCQs)
Q1: The position-time graph of an object moving with uniform velocity is:
A) A curved line B) A straight line parallel to the x-axis C) A straight line with a constant slope D) A zig-zag line
