Question

The area under acceleration versus time graph for any time interval represents... (A) Initial velocity (B) final velocity (C) change in velocity in the time interval (D) Distance covered by the particle

Short answer

The area under an acceleration versus time graph for any time interval represents the change in velocity in the time interval. This is because the area can be expressed as the product of the base (t) and the height (a), which can be rewritten as \(v - v0 = at\), illustrating the relationship between the area and the change in velocity.

Step-by-step solution

Recall basic equations of motion

Recall the following equation, which describes the relationship between acceleration (a), initial velocity (v0), final velocity (v), and time interval (t): \(v = v0 + at\)

Relationship between area and velocity

To understand the meaning of the area under the acceleration-time graph, let's consider the area of a basic shape, such as a rectangle. If the acceleration is constant throughout the time interval, the area of the rectangle is given by: Area = base × height = t × a Now, let's rewrite this in terms of velocity using the basic equation of motion: \(v - v0 = at\)

Interpret the area

By comparing the formula for the area under the acceleration-time graph with the equation of motion, we notice that the area represents the difference between the final velocity (v) and initial velocity (v0), which is the change in velocity during the time interval (t). Therefore, the correct answer is: (C) change in velocity in the time interval

Key concepts

Change in Velocity

When we talk about the change in velocity, we are essentially discussing how the speed of an object varies as it moves. This change can be influenced by the forces acting upon the object and how these forces alter over time. An acceleration-time graph is a powerful visual tool that can help us understand how an object's velocity changes.
The acceleration-time graph represents acceleration on the y-axis and time on the x-axis. It's essential to know that acceleration is how quickly velocity is changing. When looking for the "change in velocity" over a time period, we'll focus on the area under this graph for the given time interval.

• This area gives us the total change in velocity.
• A positive area indicates an increase, while a negative area shows a decrease in velocity.

The shape of the area might vary but learning to calculate or visually gauge this area is crucial for understanding the object’s change in velocity.

Equations of Motion

The equations of motion describe the relationship between key variables: time (t), acceleration (a), initial velocity (\(v_0\)), and final velocity (\(v\)). By understanding these relationships, we can predict how an object will move when subjected to constant acceleration.
The crucial equation used for understanding the area under the acceleration-time graph is:\[v = v_0 + at\] This equation relates the initial and final velocity, where:

• \(v_0\): Initial velocity of the object before acceleration.
• a: Constant acceleration the object experiences.
• t: Time interval over which the object accelerates.

By rearranging the formula, we can easily determine the change in velocity: \[v - v_0 = at\] This simple equation serves as the basis for understanding how objects respond to forces over time. It forms the core link between what we see as the area under the curve and its physical meaning.

Area Under the Curve

In physics, the area under a curve on a graph often has a special meaning. For acceleration-time graphs, the area under the curve represents the change in the object's velocity during the specified time interval. This is a fundamental concept that helps link graphical analysis to real-world motion.
Visualizing the graph as a series of geometric shapes can make it easier to understand and analyze. For example, when acceleration is constant:

• The area under the curve can be visualized as a rectangle.
• Here, the 'base' of the rectangle corresponds to the time interval (\(t\)) and the 'height' to the acceleration (\(a\)).

The mathematical representation of this is:\[\text{Area} = a \times t\] This area is equivalent to the term \(at\) in our equation of motion, directly relating the graphical concept to the physical change in velocity.
For non-constant acceleration, one might encounter more complex shapes. Yet, the principle remains: the total area still signifies the overall change in velocity.