What does the gradient of a distance-time graph represent?

The gradient represents the speed of the object. A steeper gradient indicates a higher speed, while a gradient of zero (horizontal line) indicates the object is stationary.

How do you determine acceleration from a velocity-time graph?

Acceleration is determined by calculating the gradient of the velocity-time graph. This is done using the formula $\text{acceleration} = \frac{\Delta v}{\Delta t}$, which corresponds to the 'rise over run' of the line.

What is the physical meaning of the area under a velocity-time graph?

The area under the graph represents the displacement (or distance traveled) by the object. It is calculated by finding the geometric area between the plotted line and the time axis.

How does a horizontal line differ in meaning between a distance-time and a velocity-time graph?

On a distance-time graph, a horizontal line means the object is stationary (speed is zero). On a velocity-time graph, a horizontal line means the object is moving at a constant velocity (acceleration is zero).

What does a curved line with an increasing gradient on a distance-time graph indicate?

It indicates that the object's speed is increasing over time, meaning the object is accelerating. The steeper the curve becomes, the faster the object is moving.

When calculating displacement from a velocity-time graph, how do you handle sections below the x-axis?

Areas below the x-axis represent negative displacement (moving in the opposite direction). To find total displacement, subtract these areas from the areas above the axis; to find total distance, add the absolute values of all areas.

What is a common error when calculating the gradient of a curved motion graph?

A common error is trying to use two distant points on the curve itself. Instead, you must draw a tangent line at the specific point of interest and calculate the gradient of that straight tangent line.

Why is it incorrect to calculate the area under a distance-time graph to find a physical quantity?

The area under a distance-time graph (units of $m \cdot s$) does not correspond to any standard physical kinematic quantity. Only the gradient of a distance-time graph provides useful information (speed).

What does a straight, downward-sloping line on a velocity-time graph represent?

It represents constant deceleration (or constant negative acceleration). The object's velocity is decreasing at a steady, uniform rate over time.

How can you tell if an object has returned to its starting position using a velocity-time graph?

The object has returned to its starting position when the total area above the time axis (positive displacement) is exactly equal to the total area below the time axis (negative displacement), resulting in a net displacement of zero.

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Graphs of Motion

Summary

Graphs of motion provide a visual representation of how an object's position and velocity change over time. By analyzing the geometric properties of these graphs—specifically the gradient and the area under the curve—one can derive fundamental kinematic quantities such as speed, velocity, acceleration, and displacement.

1. Definition & Core Concepts

Distance-Time Graphs ( $s-t$ ): These plots show how the total distance covered by an object changes over time. They are primarily used to determine the speed of an object at any given moment.
Velocity-Time Graphs ( $v-t$ ): These plots illustrate how the velocity of an object changes over time. They provide information about both the acceleration of the object and the total displacement achieved.
Key Variables: Time ( $t$ ) is always plotted on the horizontal x-axis as the independent variable, while distance ( $s$ ) or velocity ( $v$ ) is plotted on the vertical y-axis as the dependent variable.

2. Underlying Principles: Gradient and Area

A velocity-time graph showing a sloped line for constant acceleration and a horizontal line for constant velocity, with the shaded area below representing displacement.

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips