Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(c) Use geometry to find the displacement of the object between t = 2 and t = 5.

Verified step by step guidance

1. Step 1: Understand the problem. The displacement of the object between t = 2 and t = 5 can be found by calculating the area under the velocity graph between these time intervals. The graph consists of geometric shapes (rectangles and triangles) that can be used to compute the area.
2. Step 2: Identify the shapes under the graph between t = 2 and t = 5. From t = 2 to t = 3, the graph forms a rectangle. From t = 3 to t = 5, the graph forms a triangle.
3. Step 3: Calculate the area of the rectangle from t = 2 to t = 3. The height of the rectangle is the velocity at t = 2, which is 20 m/s, and the width is the time interval, which is 1 second. Use the formula for the area of a rectangle: $A=l\times w$.
4. Step 4: Calculate the area of the triangle from t = 3 to t = 5. The base of the triangle is the time interval, which is 2 seconds, and the height is the change in velocity, which is 20 m/s - 10 m/s = 10 m/s. Use the formula for the area of a triangle: $A=\frac{b\times h}{2}$.
5. Step 5: Add the areas of the rectangle and triangle to find the total displacement of the object between t = 2 and t = 5. The sum of these areas represents the total displacement in meters.

Key Concepts

Velocity Function

The velocity function describes how the velocity of an object changes over time. It is a graphical representation where the y-axis represents velocity (in meters per second) and the x-axis represents time (in seconds). Understanding this function is crucial for analyzing the motion of the object, as it provides insights into how fast and in what direction the object is moving at any given moment.

Displacement

Displacement is the overall change in position of an object over a specific time interval. It can be calculated by finding the area under the velocity graph between two time points. In this case, the displacement from t = 2 to t = 5 seconds can be determined by calculating the area of the geometric shapes formed by the velocity graph within that interval, which represents the total distance traveled in that time frame.

Area Under the Curve

In calculus, the area under a curve on a graph represents the integral of a function over a specified interval. For a velocity graph, this area corresponds to the displacement of the object. By using geometric shapes such as rectangles and triangles, one can calculate the area under the curve between two time points, which provides the total displacement of the object during that time period.