Question

interpreting a position vs. time graph
position vs time
use the graph to determine the velocity of the object.
what is the instantaneous velocity of the object at 3 seconds?
______ m/s
what is the average velocity of the object?
______ m/s

Explanation:

Part 1: Instantaneous Velocity at 3 seconds

Step1: Recall velocity formula from graph

Velocity is the slope of the position - time graph. For the interval from \(t = 0\) to \(t=4\) seconds, the graph is a straight line (constant velocity). The formula for slope (velocity) is \(v=\frac{\Delta x}{\Delta t}\), where \(\Delta x\) is the change in position and \(\Delta t\) is the change in time.
From the graph, at \(t = 0\), \(x = 0\) m; at \(t = 4\) s, \(x=12\) m. So \(\Delta x=12 - 0=12\) m and \(\Delta t = 4-0 = 4\) s.

Step2: Calculate velocity for \(0 - 4\) s

Using the formula \(v=\frac{\Delta x}{\Delta t}\), we substitute the values: \(v=\frac{12}{4}=3\) m/s. Since at \(t = 3\) s, the object is in the \(0 - 4\) s interval (where velocity is constant), the instantaneous velocity at \(t = 3\) s is equal to the velocity in this interval.

Part 2: Average Velocity

Step1: Recall average velocity formula

Average velocity is given by \(v_{avg}=\frac{\text{Total displacement}}{\text{Total time}}\).

Step2: Determine total displacement and total time

From the graph, the initial position \(x_i = 0\) m (at \(t = 0\) s) and the final position \(x_f=0\) m (at \(t = 12\) s). So the total displacement \(\Delta x_{total}=x_f - x_i=0 - 0 = 0\) m. The total time \(\Delta t_{total}=12 - 0=12\) s.

Step3: Calculate average velocity

Using the formula \(v_{avg}=\frac{\Delta x_{total}}{\Delta t_{total}}\), we substitute the values: \(v_{avg}=\frac{0}{12}=0\) m/s.

Answer:

Instantaneous velocity at 3 seconds: \(\boldsymbol{3}\) m/s
Average velocity: \(\boldsymbol{0}\) m/s