In this problem, we are given that a particle traverses 3/4th of the circle of radius R and we need to calculate the magnitude of the average velocity of the particle in that time.


The formula for average velocity is given by:
Average velocity = Total displacement / Total time taken


Let us assume that the circle is centered at point O and the particle starts from point A and ends at point B. Let the time taken by the particle to travel from A to B be t seconds.

We know that the circumference of a circle is given by 2πR. Therefore, the total distance traveled by the particle is (3/4) * 2πR = (3/2)πR.


The displacement of the particle is the straight line distance between points A and B. Let us assume that the angle subtended by the arc AB at the center O of the circle is θ.

The length of the arc AB is given by (θ/360) * 2πR. We know that θ = (3/4) * 360 = 270 degrees.

Therefore, the length of arc AB = (270/360) * 2πR = (3/4) * 2πR.

Using the cosine rule, we can calculate the distance AB as:

AB^2 = OA^2 + OB^2 - 2*OA*OB*cos(θ)

Since OA = OB = R, we get:

AB^2 = 2R^2 - 2R^2*cos(θ)

Substituting the value of θ, we get:

AB^2 = 2R^2 - 2R^2*cos(270)

AB^2 = 2R^2 + 2R^2 = 4R^2

Therefore, AB = 2R.

The displacement of the particle is equal to the distance AB, which is 2R.


The time taken by the particle to travel from A to B is given as t seconds.


Using the formula for average velocity, we get:

Average velocity = Total displacement / Total time taken
Average velocity = 2R / t

Therefore, the magnitude of the average velocity of the particle is 2R/t.


In this problem, we calculated the magnitude of the average velocity of a particle that traversed 3/4th of the circle of radius R. We used the formula for average velocity and calculated the displacement and time taken by the particle. Finally, we obtained the magnitude of the average velocity as 2R/t.