Understanding Motion from Graphs

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

c. When does the particle move at its greatest speed?

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Examine the graph of velocity v = f(t) over time t. The graph shows how the velocity of the particle changes with time.

Identify the points on the graph where the velocity reaches its maximum and minimum values. The greatest speed corresponds to the maximum absolute value of velocity, regardless of direction.

Observe that the graph has peaks and valleys. The highest peak or lowest valley indicates the greatest speed.

Determine the time at which these maximum or minimum values occur by looking at the horizontal axis (time) corresponding to these points.

Conclude that the particle moves at its greatest speed at the time where the graph reaches the highest or lowest point, which is the maximum absolute value of velocity.

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주요 개념

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Velocity

Velocity is a vector quantity that refers to the rate of change of an object's position with respect to time. In the context of the graph, the velocity function v = f(t) indicates how fast and in which direction the particle is moving at any given time t. Positive values indicate motion in one direction, while negative values indicate motion in the opposite direction.

Speed

Speed is the magnitude of velocity and is a scalar quantity, meaning it only has a value without direction. The greatest speed of the particle corresponds to the maximum absolute value of the velocity function on the graph. This can be determined by identifying the highest peaks and lowest troughs of the graph, as these points represent the fastest movement of the particle.

Graph Interpretation

Interpreting graphs is crucial in calculus, especially when analyzing motion. The graph of velocity versus time allows us to visualize how the particle's speed changes over time. By examining the shape of the graph, we can identify intervals of increasing or decreasing speed, as well as moments when the particle is at rest (when velocity equals zero), which aids in determining when the particle moves at its greatest speed.

Understanding Motion from GraphsThe accompanying figure shows the velocity v = f(t) of a particle moving on a horizontal coordinate line.c. When does the particle move at its greatest speed?