Prove Slope of Position Time Graph Represents Velocity, MCQs, Worksheet, Question Paper

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
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

Correct Answer: C

Explanation: A uniform velocity means the object moves equal distances in equal intervals of time, producing a straight-line graph.

Q2: What happens when the slope of the position-time graph is negative?

A) The object is moving forward
B) The object is moving backward
C) The object is stationary
D) The object is accelerating

Correct Answer: B

Explanation: A negative slope means the position decreases with time, indicating motion in the opposite direction.


Do You Know?

  • The area under a velocity-time graph gives displacement, whereas the slope of a position-time graph gives velocity.
  • If the slope of the position-time graph is zero, the object is at rest.
  • A concave-upward position-time graph indicates positive acceleration.
  • A concave-downward position-time graph indicates negative acceleration (deceleration).

Worksheet

Q1. Draw and explain the position-time graph for:

  • A body moving with uniform speed.
  • A body at rest.
  • A body undergoing acceleration.

Q2. A car starts from rest and moves with a constant velocity of 1010 m/s for 55 seconds. Draw its position-time graph and calculate its displacement.

Q3. If the position-time graph of an object is a curve, what can you say about its acceleration?


Question Paper (Total Marks: 10)

  1. Define a position-time graph and explain its importance. (2 marks)
  2. How does the slope of a position-time graph relate to velocity? (2 marks)
  3. A position-time graph has a constant positive slope. What does this indicate? (2 marks)
  4. If a position-time graph is a straight line parallel to the time axis, what does it mean? (2 marks)
  5. A body starts from rest and moves with an increasing slope on the position-time graph. What does this indicate? (2 marks)

Important Points for Quick Revision

  • The slope of the position-time graph gives velocity.
  • A straight-line graph indicates uniform motion.
  • A horizontal line means the object is at rest.
  • A curved graph indicates accelerated or decelerated motion.
  • The steeper the slope, the greater the velocity.
  • A negative slope indicates motion in the opposite direction.

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