7.1 Distance-Time Graphs
Interpretation of Distance-Time Graphs
A distance-time graph represents the displacement or distance traveled by an object over a period of time. Time is plotted on the horizontal axis ($x$-axis) and distance is plotted on the vertical axis ($y$-axis).
Gradient and Speed
- Gradient of Distance-Time Graph: Represents the speed of the object.
$$\text{Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \frac{s_2 - s_1}{t_2 - t_1}$$
- Positive Gradient: The object is moving forward or away from the starting point at a constant speed.
- Zero Gradient (Horizontal Line): The object is stationary (at rest); distance does not change as time increases.
- Negative Gradient: The object is returning to its starting point or moving in the opposite direction.
Average Speed
The total distance covered divided by the total time taken for the whole journey:
$$\text{Average Speed} = \frac{\text{Total Distance Covered}}{\text{Total Time Taken}}$$
7.2 Speed-Time Graphs
Interpretation of Speed-Time Graphs
A speed-time graph represents the speed of a moving object over time. Time is plotted on the horizontal axis ($x$-axis) and speed is plotted on the vertical axis ($y$-axis).
Gradient and Acceleration
- Gradient of Speed-Time Graph: Represents the rate of change of speed or acceleration.
$$\text{Acceleration} = \frac{\text{Change in Speed}}{\text{Change in Time}} = \frac{v - u}{t}$$
- Positive Gradient: The speed is increasing (acceleration).
- Zero Gradient (Horizontal Line): The object moves at a uniform (constant) speed; acceleration is zero ($0\text{ m s}^{-2}$).
- Negative Gradient: The speed is decreasing (deceleration or retardation).
Area Under the Graph and Distance Traveled
The area under a speed-time graph represents the total distance traveled by the object over a specific time interval.
- Rectangle Area: $\text{Length} \times \text{Width}$
- Triangle Area: $\frac{1}{2} \times \text{Base} \times \text{Height}$
- Trapezium Area: $\frac{1}{2} \times (\text{Sum of Parallel Sides}) \times \text{Height}$
Summary Comparison
| Graph Type |
Gradient Represents |
Area Under Graph Represents |
| Distance-Time |
Speed |
No physical meaning |
| Speed-Time |
Acceleration / Rate of change of speed |
Distance Traveled |