A public transportation bus travels back and forth along a road, following the position-time graph over the time period shown below. There are five points labeled on the graph. What is the average speed and velocity (magnitude and direction) of the bus between point 1 and point 4?

To find the average speed and velocity of the bus between point 1 and point 4, we need to analyze the position-time graph given.

First, let's understand the difference between speed and velocity. Speed is a scalar quantity that represents how fast an object is moving, while velocity is a vector quantity that includes both speed and direction.

Here's how we can calculate the average speed:

1. Take note of the time it takes for the bus to travel from point 1 to point 4. In this case, it takes 6 seconds.
2. Look at the position values on the graph for point 1 and point 4. The position at point 1 is 0 km, and the position at point 4 is 8 km.
3. Calculate the distance traveled by subtracting the initial position from the final position: 8 km - 0 km = 8 km.
4. Divide the distance by the time it takes: 8 km / 6 s = 1.33 km/s.

Therefore, the average speed of the bus between point 1 and point 4 is 1.33 km/s.

Now, let's calculate the average velocity:

1. Determine the displacement of the bus by subtracting the initial position from the final position: 8 km - 0 km = 8 km.
2. Divide the displacement by the time it takes: 8 km / 6 s = 1.33 km/s.

Since displacement and distance traveled are the same in this case, the average velocity of the bus between point 1 and point 4 is 1.33 km/s. However, unlike speed, velocity is a vector quantity, so we also need to specify the direction.

The direction can be determined by considering the sign of the displacement. In this case, since the displacement is positive (+8 km), the average velocity is 1.33 km/s in the positive direction.

Therefore, the average speed is 1.33 km/s, and the average velocity is 1.33 km/s in the positive direction.

To calculate the average speed and velocity of the bus between point 1 and point 4, we need to determine the displacement and time taken.

Step 1: Find the displacement:

- The displacement is the change in position from point 1 to point 4.
- Looking at the graph, we can determine the y-coordinate (position) at these two points.
- Point 1 has a position of +100 meters, and point 4 has a position of -200 meters.
- So, the displacement is: -200 - (+100) = -200 - 100 = -300 meters (negative because the bus moved in the opposite direction).

Step 2: Find the time taken:

- The time taken can be determined by looking at the x-coordinate (time) between point 1 and point 4.
- From the graph, point 1 is at 0 seconds, and point 4 is at 10 seconds.
- So, the time taken is: 10 seconds - 0 seconds = 10 seconds.

Step 3: Calculate average speed:

- Average speed is given by the formula: average speed = total distance / total time.
- In this case, the total distance is the absolute value of the displacement, as speed is a scalar quantity.
- So, the average speed is: |(-300 meters)| / 10 seconds = 300 meters / 10 seconds = 30 meters per second.

Step 4: Calculate average velocity:

- Average velocity is given by the formula: average velocity = displacement / total time.
- In this case, the displacement is already determined as -300 meters, and the total time is 10 seconds.
- So, the average velocity is: -300 meters / 10 seconds = -30 meters per second.

Conclusion:

The average speed between point 1 and point 4 is 30 meters per second, and the average velocity is -30 meters per second (in the opposite direction).

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