In this section, we delve into the graphical analysis of motion, focusing on displacement-time and velocity-time graphs. These graphs provide a visual representation of motion, helping us understand how objects move over time.
Displacement-Time Graph
The displacement of an object is defined as its distance from the initial point in a specific direction. On a displacement-time graph, displacement is the dependent variable on the y-axis, while time is the independent variable on the x-axis. These graphs are also known as position-time graphs. They can depict three different scenarios:
1. Stationary Object: The graph shows a horizontal line, indicating that the object's position does not change over time. The slope is zero, signifying that the object's velocity is zero.
2. Constant Velocity: The graph is a straight line with a constant positive slope. This indicates that the object moves at a steady rate in a specific direction.
3. Constant Acceleration: The graph shows a curve, where the slope increases with time, indicating that the object's velocity is increasing at a constant rate.
The slope of a displacement-time graph is calculated as follows:
Key takeaways from the displacement-time graph include:
- The slope represents velocity.
- A constant velocity is depicted by a straight line, whereas acceleration is shown by curved lines.
- A positive slope indicates motion in a positive direction, a negative slope indicates motion in a negative direction, and a zero slope indicates that the object is at rest.
Image courtesy of BYJUS
Example 1: Displacement-Time Graph Problem
A car starts from rest and moves in a straight line with a constant acceleration of for seconds. Plot the displacement-time graph and calculate the total displacement of the car during this period.
Solution:
1. Calculate Total Displacement:
Use the formula for displacement with constant acceleration, , where (initial velocity), (acceleration), and (time).
2. Plot the Graph:
The displacement-time graph will start at the origin (since displacement is zero at ) and curve upwards, reflecting constant acceleration. At , the displacement is .
Velocity-Time Graph
In a velocity-time graph, velocity is the dependent variable on the y-axis, and time is the independent variable on the x-axis. The slope of this graph is calculated as:
This slope represents acceleration, leading to the following conclusions:
- A steep slope indicates a rapid change in velocity.
- A shallow slope indicates a slow change in velocity.
- A negative slope indicates that the object is decelerating.
- A positive slope indicates that the object is accelerating.
- The area under the graph represents the total displacement of the object during the given time period.
Image courtesy of BYJUS
Example 2: Velocity-Time Graph Problem
A cyclist moves with a constant velocity of for seconds, then accelerates at for the next seconds. Plot the velocity-time graph and calculate the total displacement during these seconds.
Solution:
1. Calculate Displacement for Each Phase:
- Constant Velocity Phase (First (4) seconds):
Displacement is .
- Acceleration Phase (Next (2) seconds):
Use where , , and .
2. Calculate Total Displacement:
3. Plot the Graph:
The velocity-time graph shows a horizontal line at for the first seconds, then a straight line sloping upwards for the next seconds, reflecting the acceleration.