• Displacement: Displacement is a vector quantity that refers to the change in position of an object. It is measured from a specific starting point (often called the origin) and includes both magnitude and direction.
• Time: Time is plotted on the horizontal axis ($x$-axis), and displacement is plotted on the vertical axis ($y$-axis).

• Gradient (Slope):
• The gradient of the graph at any point represents the velocity of the object at that point.
• A steeper gradient indicates a higher velocity.
• A positive gradient means the object is moving in the positive direction (away from the origin).
• A negative gradient means the object is moving in the opposite direction (towards the origin).

• Shape of the Graph:
• Straight Line with Positive Slope: Indicates constant velocity in the positive direction.
• Straight Line with Negative Slope: Indicates constant velocity in the negative direction.
• Horizontal Line: Indicates that the object is stationary (zero velocity).
• Curved Line: Indicates changing velocity (i.e., acceleration or deceleration).

• Constant Velocity:
• If the graph is a straight line with a constant slope, the object is moving with a constant velocity.
• The equation of motion can be written as:

$s(t) = vt + s_0$

where $s(t)$ is the displacement at time $t$ , $v$ is the velocity, and $s_0$ is the initial displacement.

• Acceleration and Deceleration:
• If the graph is a curve, the gradient is changing, which means the velocity is changing.
• Positive Curvature (Concave Up): The object is accelerating.
• Negative Curvature (Concave Down): The object is decelerating.
• Stationary Object:
• If the graph is a horizontal line, the displacement is not changing over time, indicating that the object is stationary.

• Example: A car moving at a constant speed on a straight road.

• Example: A ball rolling down an incline, gaining speed.

• Example: A car coming to a stop.

• Example: A parked car.

• Instantaneous Velocity:
• To find the instantaneous velocity at a particular point, you calculate the gradient of the tangent to the curve at that point.
• Average Velocity:
• The average velocity over a time interval can be found by taking the gradient of the straight line connecting the initial and final points on the graph.
• Formula:

$\text{Average velocity} = \frac{\text{Change in displacement}}{\text{Change in time}} = \frac{s_2 - s_1}{t_2 - t_1}$

• Direction of Motion:
• If the graph returns to the $x$-axis (displacement = $0$), the object has returned to its starting point.
• A graph crossing the $x$-axis indicates a change in the direction of motion.
• Non-Linear Graphs:
• A non-linear graph suggests that the object is undergoing acceleration or deceleration.
• The steeper the curve, the greater the rate of acceleration or deceleration.

Displacement-time graphs are a visual way to represent and analyse an object's motion. The slope of the graph provides information about the object's velocity, while the shape of the graph indicates whether the object is moving at a constant velocity, accelerating, or decelerating. Understanding how to interpret these graphs is crucial for solving mechanics problems related to motion.