Question

Slope of the velocity-time graph gives of a moving body. (A) displacement (B) acceleration (C) initial velocity (D) final velocity

Short answer

The slope of the velocity-time graph represents (B) acceleration, as it is defined as the rate at which velocity changes with respect to time.

Step-by-step solution

Understanding the velocity-time graph

The velocity-time graph is a graphical representation of the motion of a moving object that shows the object's velocity with respect to time. The graph has time represented on the x-axis while the instantaneous velocity is represented on the y-axis.

Calculating the slope of the graph

The slope of the graph is the rate at which one variable is changing with respect to another. In the case of a velocity-time graph, the slope represents the rate at which the velocity is changing with respect to time. Mathematically, the slope of the graph can be represented as: Slope = \(\frac{change\:in\:velocity}{change\:in\:time}\)

Relating the slope to the options provided

Now, we need to determine which concept from kinematics (displacement, acceleration, initial velocity, or final velocity) is represented by the slope of the velocity-time graph. Let's examine each option: (A) The slope of the graph cannot represent displacement, as displacement is the change in position, not the rate of change of velocity with respect to time. (B) Acceleration is defined as the rate at which velocity changes with respect to time. Therefore, the slope of the velocity-time graph can be represented as acceleration. Mathematically, acceleration is defined as: Acceleration = \(\frac{change\:in\:velocity}{change\:in\:time}\) (C) Initial velocity is the velocity of the body at the start of the time interval or when time is zero. It is not related to the slope of the graph, which represents the rate of change of velocity with respect to time. (D) Final velocity is the velocity of the body at the end of the time interval or when the motion stops. It also does not represent the slope of the graph, which indicates the rate of change of velocity with respect to time. Based on the above analysis, we can determine that:

Conclusion

The slope of the velocity-time graph represents (B) acceleration.

Key concepts

Understanding Slope Calculation

Slope calculation is a core concept in mathematics and physics, particularly when analyzing graphs. On a velocity-time graph, the slope gives insight into how the velocity of an object changes over time. Calculating the slope involves determining how one quantity changes in relation to another. Here, we're focused on how velocity changes with respect to time. Essentially, slope is calculated using the formula:

• Slope = \( \frac{ \text{change in velocity} }{ \text{change in time} } \)

This value represents the rate of change of velocity, which is a fundamental aspect of analyzing motion. It provides a visual and mathematical way to understand how fast the velocity of an object changes, which directly relates to the object’s acceleration. By interpreting this slope correctly, particularly on a velocity-time graph, you can predict the behavior of moving objects.

Basics of Kinematics in Context

Kinematics is the branch of physics that deals with motion without considering the forces that cause it. It's the study of how positions, velocities, and accelerations of objects change over time. Within kinematics, there are several fundamental concepts:

• Displacement: This is the change in position of an object. Unlike distance, which measures how much ground an object covers, displacement considers the overall change in position.
• Velocity: Velocity is a vector quantity referring to the speed of an object in a certain direction. It's different from speed, which is scalar and doesn't account for direction.
• Acceleration: This refers to how the velocity of an object changes over time.

Understanding how these quantities relate and change is crucial for deciphering motion through graphs. For instance, the slope of a velocity-time graph, as calculated earlier, gives us direct information about acceleration.

Exploring Acceleration and Its Significance

Acceleration is a key concept in kinematics, describing how an object's velocity changes over time. On a velocity-time graph, acceleration is directly represented by the slope. Here's why it matters:

• Positive acceleration: When the slope is positive, the object is speeding up. This means its velocity is increasing over time.
• Negative acceleration (deceleration): A negative slope indicates the object is slowing down, as its velocity is decreasing.
• Constant acceleration: A straight line on the graph shows constant acceleration, meaning the velocity changes at a steady rate.

Understanding acceleration helps in predicting future motion and analyzing forces in advanced studies. It's essential for real-world applications, such as determining the forces exerted on a car as it speeds up or slows down. Acceleration provides a way to quantify how an object moves and interacts with its environment, making it an indispensable concept in both physics and engineering.