Which Graph Represents the Motion of an Object with Positive Acceleration?

Correct Answer: Option (C)

Understanding the Concept of Positive Acceleration

To identify the graph representing positive acceleration, we must first understand the relationship between position, velocity, acceleration, and time. Acceleration describes how rapidly the velocity of an object changes with time. Mathematically, acceleration is expressed as:

Acceleration = Change in Velocity / Change in Time

An object has positive acceleration when its velocity increases with time. Therefore, a velocity-time graph with a positive slope directly represents positive acceleration. The steeper the upward slope of the velocity-time graph, the greater the positive acceleration of the object.

How to Identify Acceleration from Motion Graphs

The meaning of the slope depends on the type of graph given in the question. In a position-time graph, the slope represents velocity. Therefore, a straight line with a constant slope indicates constant velocity and zero acceleration. In a velocity-time graph, however, the slope directly represents acceleration. Thus, an upward-sloping straight line on a velocity-time graph indicates positive acceleration.

Detailed Analysis of Each Option

Option (A): Position-Time Graph with a Constant Positive Slope

In option (A), position increases linearly with time. The graph is a straight line with a constant positive slope. Since the slope of a position-time graph represents velocity, a constant slope means that the object is moving with constant positive velocity.

Although the object is moving in the positive direction, its velocity is not increasing. There is no change in velocity with time, so the acceleration is zero. Therefore, option (A) does not represent positive acceleration.

Option (B): Horizontal Position-Time Graph

In option (B), the position of the object remains constant as time passes. A horizontal position-time graph has zero slope, which means that the velocity of the object is zero.

The object is at rest and its velocity is not changing. Since acceleration is the rate of change of velocity, the acceleration is also zero. Therefore, option (B) cannot represent an object with positive acceleration.

Option (C): Velocity-Time Graph with an Increasing Velocity

In option (C), velocity increases continuously with time. The velocity-time graph is a straight line with a positive upward slope. Since the slope of a velocity-time graph represents acceleration, a positive slope indicates positive acceleration.

Mathematically:

Acceleration = Slope of the Velocity-Time Graph = Δv / Δt

As time increases, the velocity of the object also increases. This clearly shows that the object is speeding up and has a positive acceleration. Therefore, option (C) is the correct answer.

Option (D): Horizontal Velocity-Time Graph

In option (D), the velocity remains constant with time. The velocity-time graph is horizontal, so its slope is zero. Because the slope of a velocity-time graph represents acceleration, a zero slope means zero acceleration.

The object is moving with constant velocity, but its velocity is not increasing or decreasing. Therefore, option (D) does not represent positive acceleration.

Why Is Option (C) the Correct Answer?

Option (C) is correct because it shows an upward-sloping velocity-time graph. The velocity of the object increases as time increases, which means there is a positive change in velocity over a positive time interval. As a result, the acceleration is positive.

The other three options represent zero acceleration. Option (A) shows constant velocity through a straight-line position-time graph, option (B) shows an object at rest, and option (D) shows constant velocity through a horizontal velocity-time graph. Only option (C) shows velocity increasing with time.

Relationship Between the Slope of a Graph and Motion

Understanding the slope of different motion graphs is essential for solving questions related to kinematics. The slope of a position-time graph gives the velocity of the object, while the slope of a velocity-time graph gives its acceleration. Therefore, students should first identify the quantities represented on the vertical and horizontal axes before interpreting any motion graph.

A straight position-time graph with constant slope represents constant velocity and zero acceleration. A horizontal position-time graph represents zero velocity. An upward-sloping velocity-time graph represents positive acceleration, while a horizontal velocity-time graph represents constant velocity and zero acceleration.

Final Answer

The correct answer is Option (C). The velocity-time graph in option (C) has a positive slope, which means that the velocity of the object increases with time. Since acceleration is the rate of change of velocity with time, this graph represents motion with positive acceleration.