Understanding Motion from Graphs


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


a. When does the particle move forward? Move backward? Speed up? Slow down?

Verified step by step guidance

1. To determine when the particle moves forward, look for intervals where the velocity v = f(t) is positive. This occurs when the graph is above the t-axis.
2. To determine when the particle moves backward, look for intervals where the velocity v = f(t) is negative. This occurs when the graph is below the t-axis.
3. To determine when the particle speeds up, identify intervals where the magnitude of velocity is increasing. This can be seen when the graph is moving away from the t-axis, either upwards or downwards.
4. To determine when the particle slows down, identify intervals where the magnitude of velocity is decreasing. This can be seen when the graph is moving towards the t-axis, either upwards or downwards.
5. Analyze the graph: From t = 0 to t = 1, the particle moves forward and slows down. From t = 1 to t = 3, it moves backward and speeds up. From t = 3 to t = 5, it moves backward and slows down. From t = 5 to t = 6, it moves forward and speeds up. From t = 6 to t = 7, it moves forward and slows down. From t = 7 to t = 9, the velocity is zero, indicating no movement.

Key Concepts

Velocity and Direction of Motion

Velocity is a vector quantity that indicates the rate of change of position with respect to time. When the velocity is positive, the particle moves forward; when negative, it moves backward. Understanding the sign of the velocity function is crucial for determining the direction of motion.

Acceleration and Speed Changes

Acceleration is the rate of change of velocity. A particle speeds up when the velocity and acceleration have the same sign (both positive or both negative) and slows down when they have opposite signs. Analyzing the graph for intervals where the velocity is increasing or decreasing helps identify these changes.

Graph Interpretation

Interpreting the graph of velocity over time involves analyzing its shape and key features, such as peaks, troughs, and zero crossings. These features indicate when the particle is at rest (velocity = 0), changing direction, or experiencing changes in speed, which are essential for answering questions about motion.