What does the area under a velocity vs. time graph represent?
What does the area under a velocity vs time graph represent? The area under a speed-time graph represents the distance travelled. This is a velocity time graph of an object moving in a straight line due North.
How to estimate the area under the graph?
When computing this area via rectangles, there are several things to know:
- What interval are we on? ...
- How many rectangles will be used? ...
- What is the width of each individual rectangle? ...
- What points will determine the height of the rectangle? ...
- What is the actual height of the rectangle? ...
- We approximate the area A with a Riemann sum A ≈ ∑ k = 1 n f ( x k ∗) Δ x .
What does the area of a velocity time graph give?
The area under velocity time graph gives displacement. This is because velocity is the first derivative of displacement. Finding area under a graph is the same as Integrating. So Integration of the first derivative of displacement is displacement itself. Likewise, the area under acceleration time graph will give you velocity.
How do you calculate velocity time graph?
Using the Slope Equation
- Pick two points on the line and determine their coordinates.
- Determine the difference in y-coordinates for these two points ( rise ).
- Determine the difference in x-coordinates for these two points ( run ).
- Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).
Calculating the Area of a Rectangle
Now we will look at a few example computations of the area for each of the above geometric shapes. First consider the calculation of the area for a few rectangles. The solution for finding the area is shown for the first example below. The shaded rectangle on the velocity-time graph has a base of 6 s and a height of 30 m/s.
Calculating the Area of a Triangle
Now we will look at a few example computations of the area for a few triangles. The solution for finding the area is shown for the first example below. The shaded triangle on the velocity-time graph has a base of 4 seconds and a height of 40 m/s.
Calculating the Area of a Trapezoid
Finally we will look at a few example computations of the area for a few trapezoids. The solution for finding the area is shown for the first example below. The shaded trapezoid on the velocity-time graph has a base of 2 seconds and heights of 10 m/s (on the left side) and 30 m/s (on the right side).
Alternative Method for Trapezoids
An alternative means of determining the area of a trapezoid involves breaking the trapezoid into a triangle and a rectangle. The areas of the triangle and rectangle can be computed individually; the area of the trapezoid is then the sum of the areas of the triangle and the rectangle. This method is illustrated in the graphic below.
Investigate!
The widget below computes the area between the line on a velocity-time plot and the axes of the plot. This area is the displacement of the object. Use the widget to explore or simply to practice a few self-made problems.
We Would Like to Suggest ..
Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives. We would like to suggest that you combine the reading of this page with the use of our Two Stage Rocket Interactive.
Calculating The Area of A Rectangle
Calculating The Area of A Triangle
- Now we will look at a few example computations of the area for a few triangles. The solution for finding the area is shown for the first example below. The shaded triangle on the velocity-time graph has a base of 4 seconds and a height of 40 m/s. Since the area of triangle is found by using the formula A = ½ * b * h, the area is ½ * (4 s) * (40 m/s...
Calculating The Area of A Trapezoid
- Finally we will look at a few example computations of the area for a few trapezoids. The solution for finding the area is shown for the first example below. The shaded trapezoid on the velocity-time graph has a base of 2 seconds and heights of 10 m/s (on the left side) and 30 m/s (on the right side). Since the area of trapezoid is found by using the formula A = ½ * (b) * (h1 + h2), the a…
Alternative Method For Trapezoids
- An alternative means of determining the area of a trapezoid involves breaking the trapezoid into a triangle and a rectangle. The areas of the triangle and rectangle can be computed individually; the area of the trapezoid is then the sum of the areas of the triangle and the rectangle. This method is illustrated in the graphic below. Triangle: Area = ½ * (2 s) * (20 m/s) = 20 m Rectangle: Area = (2 …