Question

activity a (continued from previous page)

1. slope is the steepness of a graph. to find the slope of a line, divide the change in y - value (rise) by the change in x - value (run). like velocity, slope can be positive, zero, or negative.

fill in the slope of each segment of your position - time graph, along with the runner’s velocity during each time interval, in the table below.

time interval	slope	velocity (y/s)
0 sec - 1 sec
1 sec - 2 sec
2 sec - 3 sec
3 sec - 4 sec

1. examine your velocities and the position - time graph you made. how is the slope of a position - time graph related to the velocity of the runner?
2. on the left side of the gizmo, select the velocity - time graph tab. use the green probes to compare the velocity - time graph to the position - time graph.

a. how does a velocity - time graph show that a runner is moving fast?
b. how does a velocity - time graph show that a runner is moving from left to right?

1. to the right is a position - time graph of a runner.

first, sketch what you think his velocity - time graph will look like on the blank axes at the far right.
then check your answer in the gizmo.

Explanation:

Step1: Recall slope - velocity relationship

The slope of a position - time graph gives the velocity. The formula for slope $m=\frac{\Delta y}{\Delta x}$, where in the context of a position - time graph, $\Delta y$ is the change in position and $\Delta x$ is the change in time.

Step2: Analyze velocity - time graph for fast motion

On a velocity - time graph, a runner is moving fast when the magnitude of the velocity value (the y - value of the graph) is large. Whether the velocity is positive or negative, a larger absolute value of velocity indicates faster motion.

Step3: Analyze velocity - time graph for left - to - right motion

On a velocity - time graph, a runner moving from left to right (assuming the positive x - direction is to the right) has a positive velocity value. A positive y - value on the velocity - time graph indicates motion in the positive direction (from left to right).

Step4: Sketch velocity - time graph from position - time graph

For the given position - time graph:

• From 0 - 1 second, the position is decreasing at a constant rate. So the velocity is negative and constant.
• From 1 - 2 seconds, the position is still decreasing at a constant rate, so the velocity remains negative and constant (same magnitude as 0 - 1 second assuming linear motion).
• From 2 - 3 seconds, the position is increasing at a constant rate, so the velocity is positive and constant.
• From 3 - 4 seconds, the position is increasing at a constant rate, so the velocity is positive and constant (same magnitude as 2 - 3 seconds assuming linear motion).

Answer:

1. Without the actual position - time graph data, we can't fill in the table. But in general, for each time interval, the slope (velocity) $v=\frac{\text{Change in position}}{\text{Change in time}}$.
2. The slope of a position - time graph is equal to the velocity of the runner. The steeper the slope (either positive or negative), the greater the magnitude of the velocity.
3. A. A runner is moving fast when the absolute value of the velocity (y - value on the velocity - time graph) is large.

B. A runner is moving from left to right when the velocity (y - value on the velocity - time graph) is positive.

1. Sketch a velocity - time graph with a negative constant velocity from 0 - 2 seconds and a positive constant velocity from 2 - 4 seconds. The magnitude of the positive and negative velocities should be the same if the position - time graph has the same steepness for the decreasing and increasing parts.