Creating and Analysing Speed-Time and Velocity-Time Graphs
Speed-time and velocity-time graphs are fundamental in kinematics as they provide a visual representation of an object's motion, allowing us to understand and analyse its behaviour over time.
- Speed-Time Graphs: Here, the speed of an object is plotted against time. The gradient at any point on the graph represents the acceleration of the object. The area under the graph, between the graph line and the time axis, represents the distance travelled by the object.
- Velocity-Time Graphs: These are similar to speed-time graphs, but velocity includes direction. The gradient still represents acceleration, but the area under the graph now represents displacement, which is distance taking into account the direction of travel.
Gradient in velocity-time graph
Image Courtesy Science Facts
Using Graphical Areas to Calculate Distances and Displacements
The area under a speed-time or velocity-time graph can be used to calculate the distance or displacement of an object.
- Distance: In a speed-time graph, the area under the graph (between the graph line and the time axis) represents the total distance travelled by the object. This is calculated by integrating the speed-time graph.
- Displacement: In a velocity-time graph, the area under the graph represents the object's displacement. This is similar to distance, but takes into account the direction of travel. Displacement is calculated by integrating the velocity-time graph.
Interpreting Constant and Changing Speeds in Graphical Form
Speed-time and velocity-time graphs can also be used to interpret an object's speed.
- Constant Speed: If the speed or velocity is constant, the graph will be a straight horizontal line. The gradient (slope) of the line will be zero, indicating no acceleration.
- Changing Speed: If the speed or velocity is changing, the graph will not be a straight line. The gradient of the line will indicate the object's acceleration. A positive gradient indicates increasing speed (acceleration), while a negative gradient indicates decreasing speed (deceleration).
Velocity-time graph
Image Courtesy CK12
Practical Examples
Consider a velocity-time graph where an object accelerates from rest to a velocity of 10 m/s over 5 seconds. The gradient of the line would be 2 m/s² (acceleration), and the area under the graph would represent the displacement of the object (½ x base x height = ½ x 5 s x 10 m/s = 25 m).
FAQ
If an object changes direction during its motion, the velocity-time graph will cross the time axis. The total distance travelled by the object is the sum of the absolute values of the areas of the regions above and below the time axis. The area of each region can be calculated as the product of the magnitude of the velocity (height of the region) and the time interval (width of the region).
If the velocity-time graph is a horizontal line above the time axis, it means that the object is moving at a constant velocity in the positive direction. The height of the line above the time axis represents the magnitude of the velocity. The object is not accelerating or decelerating because its velocity is not changing over time.
If the speed-time graph is a straight line sloping downwards, it means that the object is decelerating, or slowing down. The slope of the line gives the rate of deceleration. If the line crosses the time axis and continues downwards, it means that the object has changed direction and is now speeding up in the opposite direction. The area under the graph before it crosses the time axis represents the distance travelled in the original direction, and the area under the graph after it crosses the time axis represents the distance travelled in the opposite direction.
A velocity-time graph not only shows the speed of an object but also its direction of motion. If the velocity is positive, the object is moving in the positive direction, and if the velocity is negative, the object is moving in the negative direction. The direction could be up or down, left or right, or any other pair of opposite directions, depending on the context. If the graph crosses the time axis, this indicates a change in direction.
A distance-time graph and a speed-time graph both represent motion, but they do so in different ways. A distance-time graph shows how the position of an object changes over time. The slope of a distance-time graph at any point gives the speed of the object at that moment. On the other hand, a speed-time graph shows how the speed of an object changes over time. The slope of a speed-time graph at any point gives the acceleration of the object at that moment. The area under the graph in a speed-time graph represents the distance travelled by the object.
Practice Questions
The velocity-time graph would consist of three sections: an upward sloping line from (0,0) to (10,20) representing the acceleration, a horizontal line from (10,20) to (20,20) representing uniform speed, and a downward sloping line from (20,20) to (25,0) representing deceleration. The total displacement can be calculated as the area under the graph. This is the sum of the areas of a triangle (1/2 10 s 20 m/s = 100 m), a rectangle (10 s 20 m/s = 200 m), and another triangle (1/2 5 s * 20 m/s = 50 m). So, the total displacement is 100 m + 200 m + 50 m = 350 m.
The acceleration can be calculated from the gradient of the velocity-time graph. As the line slopes downwards, the acceleration is negative. The change in velocity is -15 m/s over a time period of 3 s, so the acceleration is -15 m/s / 3 s = -5 m/s². The displacement can be calculated as the area under the graph, which is the area of a triangle: 1/2 base height = 1/2 3 s 15 m/s = 22.5 m.
