Question

use the graph to determine the instantaneous speed at 2 seconds. m/s what is the average speed over the entire trip? m/s

Explanation:

Part 1: Instantaneous Speed at 2 seconds

Step1: Understand the graph type

The graph is a position - time graph. In a position - time graph, the slope of the graph at a point gives the instantaneous speed. For a linear segment of the graph, the slope is constant, so the instantaneous speed during that linear segment is equal to the slope.

Step2: Identify the segment at t = 2s

At t = 2s, the object is on the linear segment from t = 0s to t = 4s (since the graph is a straight line from (0,0) to (4,12)).

Step3: Calculate the slope (instantaneous speed)

The formula for the slope (speed) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(v=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}\).
For the points \((0,0)\) and \((4,12)\), \(\Delta y=12 - 0 = 12\) m and \(\Delta x = 4-0=4\) s.
So the slope (speed) \(v=\frac{12}{4}=3\) m/s. Since the segment from 0 to 4 seconds is linear, the instantaneous speed at t = 2s is also 3 m/s.

Part 2: Average Speed over the entire trip

Step1: Recall the formula for average speed

The formula for average speed is \(v_{avg}=\frac{\text{total distance traveled}}{\text{total time taken}}\).

Step2: Determine total distance and total time

• The object moves from position 0 to 12 m (distance of 12 m) and then back from 12 m to 0 m (another distance of 12 m). So the total distance traveled \(d = 12+12=24\) m.
• The total time taken from t = 0s to t = 12s, so \(t = 12\) s.

Step3: Calculate average speed

Using the formula \(v_{avg}=\frac{d}{t}\), we substitute \(d = 24\) m and \(t = 12\) s.
\(v_{avg}=\frac{24}{12}=2\) m/s.

Answer:

Instantaneous speed at 2 seconds: \(\boldsymbol{3}\) m/s.
Average speed over the entire trip: \(\boldsymbol{2}\) m/s.