In such questions, North = +j
South = -j
East = +i
West = -i
By using these you can find direction in this question.

The Problem

A particle is initially moving towards the east with a velocity of 5m/s. After 10 seconds, its direction changes towards the north with the same velocity. We are required to find the average acceleration of the particle.

Solution

To find the average acceleration of the particle, we need to determine the change in velocity and the time taken for this change.

Step 1: Determine the change in velocity

When the particle changes its direction from east to north, its velocity will change. However, the magnitude of the velocity remains the same (5m/s). To determine the change in velocity, we can break it into its horizontal and vertical components.

The initial velocity towards the east can be represented as (5m/s, 0m/s), where the first component represents the velocity in the x-direction (east) and the second component represents the velocity in the y-direction (north).

When the direction changes towards the north, the velocity becomes (0m/s, 5m/s), where the first component represents the velocity in the x-direction (east) and the second component represents the velocity in the y-direction (north).

The change in velocity can be calculated by subtracting the initial velocity from the final velocity:

Change in velocity = Final velocity - Initial velocity
= (0m/s, 5m/s) - (5m/s, 0m/s)
= (-5m/s, 5m/s)

Step 2: Determine the time taken

The time taken for the change in velocity is given as 10 seconds.

Step 3: Calculate the average acceleration

The average acceleration can be calculated using the formula:

Average acceleration = Change in velocity / Time taken

Substituting the values we obtained:

Average acceleration = (-5m/s, 5m/s) / 10s
= (-0.5m/s^2, 0.5m/s^2)

Step 4: Determine the direction of the average acceleration

The direction of the average acceleration can be determined by finding the angle it makes with the positive x-axis (east).

Using trigonometry, we can calculate the angle using the components of the average acceleration:

Angle = arctan(0.5m/s^2 / -0.5m/s^2)
= arctan(-1)
= -45 degrees

Since the angle is negative, it means the average acceleration is directed towards the southwest direction.

Therefore, the average acceleration of the particle is -0.5m/s^2 in the southwest direction.