Deceleration on a speed-time graph is represented by the gradient (slope) of the line, and its formula is the same as that for acceleration, where a negative value indicates deceleration.
The Formula for Deceleration
The formula for deceleration, which is essentially negative acceleration, calculates the rate at which an object's velocity decreases over time. It is derived from the change in velocity divided by the change in time.
The formula is:
$a = \frac{\Delta v}{\Delta t} = \frac{v - v_0}{t - t_0}$
Where:
| Symbol | Description | Unit (SI) |
|---|---|---|
| a | Acceleration/Deceleration: The rate of change of velocity. A negative value for a signifies deceleration. | m/s² |
| $\Delta v$ | Change in Velocity: The difference between the final and initial velocities. | m/s |
| $\Delta t$ | Change in Time: The duration over which the velocity change occurs. | s |
| v | Final Velocity: The velocity of the object at the end of the time interval. | m/s |
| $v_0$ | Initial Velocity: The velocity of the object at the beginning of the time interval. Often represented as $u$ in some contexts. | m/s |
| t | Final Time: The time measurement at the end of the interval. | s |
| $t_0$ | Initial Time: The time measurement at the beginning of the interval. Often, for simplicity, $t_0$ is assumed to be 0, simplifying the denominator to just t (total time taken). | s |
Understanding Deceleration on a Speed-Time Graph
On a speed-time graph, velocity is plotted on the y-axis and time on the x-axis. The gradient (slope) of the line segment represents the acceleration.
- Positive Gradient: Indicates positive acceleration (speeding up).
- Zero Gradient (Horizontal Line): Indicates constant speed (no acceleration).
- Negative Gradient (Downward Sloping Line): Indicates deceleration (slowing down). This is when the final velocity ($v$) is less than the initial velocity ($v_0$), resulting in a negative change in velocity ($\Delta v$).
Calculating Deceleration: A Practical Example
Let's consider a car that is slowing down.
- Scenario: A car traveling at 25 m/s begins to brake and slows down to 5 m/s over a period of 4 seconds.
- Identify Variables:
- Initial velocity ($v_0$) = 25 m/s
- Final velocity ($v$) = 5 m/s
- Initial time ($t_0$) = 0 s (assuming the measurement starts at this point)
- Final time ($t$) = 4 s
- Apply the Formula:
$a = \frac{v - v_0}{t - t_0}$
$a = \frac{5 \text{ m/s} - 25 \text{ m/s}}{4 \text{ s} - 0 \text{ s}}$
$a = \frac{-20 \text{ m/s}}{4 \text{ s}}$
$a = -5 \text{ m/s}^2$
The result of -5 m/s² indicates a deceleration of 5 meters per second squared.
Key Characteristics of Deceleration
- Direction: Deceleration occurs when the acceleration vector is in the opposite direction to the velocity vector.
- Impact on Speed: It always leads to a decrease in the magnitude of speed if the object is moving in a straight line.
- Graphical Representation: Visually, on a speed-time graph, a steeper negative slope indicates a greater rate of deceleration.
