Understanding Motion from Graphs


The accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.


b. When is the particle’s acceleration positive? Negative? Zero?

Verified step by step guidance

1. To determine when the particle's acceleration is positive, negative, or zero, we need to analyze the graph of velocity v = f(t) with respect to time t. Acceleration is the derivative of velocity, so we are looking for the slope of the velocity graph.
2. Acceleration is positive when the slope of the velocity graph is positive, meaning the graph is increasing. Look for intervals where the graph is rising as time progresses.
3. Acceleration is negative when the slope of the velocity graph is negative, meaning the graph is decreasing. Identify intervals where the graph is falling as time progresses.
4. Acceleration is zero when the slope of the velocity graph is zero, meaning the graph is flat or horizontal. Find intervals where the graph is constant and not changing with time.
5. Examine the graph: From t = 0 to t = 1, the graph is decreasing, indicating negative acceleration. From t = 1 to t = 2, the graph is increasing, indicating positive acceleration. From t = 2 to t = 4, the graph is flat, indicating zero acceleration. Continue this analysis for the entire graph to identify all intervals of positive, negative, and zero acceleration.

Key Concepts

Velocity and Acceleration

Velocity is the rate of change of position with respect to time, represented as the function v = f(t) in the graph. Acceleration, on the other hand, is the rate of change of velocity with respect to time. When analyzing motion, understanding how velocity changes helps determine when the acceleration is positive (velocity increasing), negative (velocity decreasing), or zero (constant velocity).

Graph Interpretation

Interpreting graphs is crucial in calculus, especially for understanding motion. The graph of velocity versus time shows how the velocity of the particle changes over time. The slope of the velocity graph indicates acceleration: a positive slope means positive acceleration, a negative slope indicates negative acceleration, and a horizontal line (zero slope) signifies zero acceleration.

Critical Points and Intervals

Critical points on the velocity graph occur where the graph changes direction, which can indicate changes in acceleration. By identifying intervals where the graph is increasing or decreasing, one can determine when the particle's acceleration is positive, negative, or zero. This analysis is essential for answering questions about the motion of the particle over time.