At one instant a bicyclist is 50 m due east of a park's flagpole, going due south with a speed of 18 m/s. Then, 13 s later, the cyclist is 45 m due north of the flagpole, going due east with a speed of 13 m/s. For the cyclist in this 13 s interval what is the displacement (m) and direction north of west, average velocity (m/s) and direction north of west and the average acceleration and direction north of east

To solve this problem, we need to break it down into different parts and calculate the displacement, velocity, and acceleration for each part.

1. Displacement:
To find the displacement, we need to determine the change in position of the bicyclist. We will consider the position of the bicyclist at the start and end of the 13-second interval.

a. From the given information, at the start, the bicyclist is 50 m due east of the flagpole.
b. After 13 seconds, the bicyclist is 45 m due north of the flagpole.

We can calculate the displacement using the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two legs.

Using the Pythagorean theorem, we find:
Displacement = √[(45)^2 + (50)^2] = √(2025 + 2500) = √4525 ≈ 67.28 m

Therefore, the displacement of the bicyclist in this 13-second interval is approximately 67.28 meters.

2. Average Velocity:
The average velocity is the displacement divided by the time taken. To find the average velocity, we need to determine the time taken by the bicyclist for the 13-second interval.

Therefore, the average velocity is given by:
Average Velocity = Displacement / Time

Given that the time is 13 seconds, we get:
Average Velocity = 67.28 m / 13 s ≈ 5.17 m/s

The average velocity of the bicyclist in this 13-second interval is approximately 5.17 m/s.

3. Direction North of West:
To determine the direction of the displacement and average velocity, we can use trigonometry.

a. For the displacement:
The angle θ can be found as:
θ = arctan(45 m / 50 m) = arctan(0.9) ≈ 41.19 degrees

Therefore, the displacement is directed at an angle of approximately 41.19 degrees north of west.

b. For the average velocity:
Since the average velocity is calculated using the displacement, it will have the same direction.

Therefore, the average velocity is directed at an angle of approximately 41.19 degrees north of west.

4. Average Acceleration:
To find the average acceleration, we will use the formula:
Average Acceleration = (Change in Velocity) / Time

To calculate the change in velocity, we will subtract the initial velocity from the final velocity.

Given that the initial velocity is 18 m/s (going south) and the final velocity is 13 m/s (going east), we get:

Change in Velocity = Final Velocity - Initial Velocity
Change in Velocity = 13 m/s - (-18 m/s)
Change in Velocity = 13 m/s + 18 m/s = 31 m/s

Given that the time is 13 seconds, we can calculate the average acceleration:

Average Acceleration = (31 m/s) / (13 s)
Average Acceleration ≈ 2.38 m/s^2

Therefore, the average acceleration of the bicyclist in this 13-second interval is approximately 2.38 m/s^2, directed north of east.

To find the displacement, average velocity, and average acceleration of the cyclist in the given interval, we'll break down the problem into steps:

Step 1: Calculate the cyclist's displacement:
- The displacement is the change in position of the cyclist from the starting point to the ending point.
- In this interval, the cyclist moves 50 m to the south and then 45 m to the north.
- The net displacement is the difference between these two movements: 50 m south - 45 m north = 5 m south.
- Therefore, the displacement is 5 m south.

Step 2: Calculate the average velocity:
- Average velocity is defined as the displacement divided by the time taken.
- Since the problem specifies a time interval of 13 seconds, we will use this value to calculate the average velocity.
- Average velocity = displacement / time.
- Average velocity = 5 m south / 13 s = approximately 0.38 m/s south.

Step 3: Determine the direction north of west:
- To specify the direction, we'll use reference axes. North will be considered positive, and west will be negative.
- Since the displacement is 5 m south, it means that it is in the opposite direction of the positive north axis.
- Therefore, the direction would be measured as 180 degrees (south) with respect to the positive north axis.

Step 4: Calculate the average acceleration:
- Average acceleration is defined as the change in velocity divided by the time taken.
- In this problem, we don't have information about the time it takes for the velocity to change. Hence, we cannot calculate the average acceleration.

Step 5: Determine the direction of the average acceleration:
- Since we are unable to calculate the average acceleration, we cannot determine its direction.

In summary, for the cyclist in this 13-second interval:
- Displacement: 5 m south
- Direction: 180 degrees (south)
- Average velocity: Approximately 0.38 m/s south
- Direction: 180 degrees (south)
- Average acceleration: Unable to determine
- Direction: Unable to determine