a particle is moving on a straight line Xx with constant velocity 20m/s.an observer B is sitting at perpendicular distance 50m from Xx.the angular velocity of particle A w.r.t B at the instant theta =45 degree

To find the angular velocity of particle A with respect to observer B when the angle θ is 45 degrees, we can use some fundamental concepts from physics, particularly those related to motion and angular measurements. Let's break this down step by step.

Understanding the Setup

We have a particle A moving along a straight line (let's call it the X-axis) with a constant velocity of 20 m/s. Observer B is positioned at a perpendicular distance of 50 meters from this line. The angle θ, which is the angle between the line connecting particle A to observer B and the X-axis, is given as 45 degrees.

Visualizing the Situation

Imagine a right triangle formed by the position of particle A, the position of observer B, and the perpendicular distance from the line of motion to observer B. The base of this triangle is the distance along the X-axis, and the height is the perpendicular distance from the X-axis to observer B, which is 50 meters.

Calculating the Angular Velocity

The angular velocity (ω) can be defined as the rate of change of the angle θ with respect to time. We can relate this to the linear velocity of the particle and the distance from the observer to the particle. The formula for angular velocity in this context is:

• ω = v / r

Where:

• v is the linear velocity of the particle (20 m/s in this case).
• r is the distance from observer B to the particle A at the moment when θ = 45 degrees.

Finding the Distance r

At θ = 45 degrees, the distance r can be calculated using the properties of a right triangle. Since θ is 45 degrees, the opposite and adjacent sides are equal. Thus, we can use the following relationship:

• r = height / sin(θ) = 50 m / sin(45°) = 50 m / (√2/2) = 50√2 m.

Substituting Values

Now we can substitute the values into the angular velocity formula:

• ω = v / r = 20 m/s / (50√2 m) = 20 / (50√2).

Calculating this gives:

• ω = 20 / (50√2) = 2 / (5√2) rad/s.

Final Result

Thus, the angular velocity of particle A with respect to observer B at the instant when θ = 45 degrees is:

• ω ≈ 0.283 rad/s (after calculating 2 / (5√2)).

This result indicates how quickly the angle between the line of sight from observer B to particle A is changing as particle A moves along the X-axis. Understanding these relationships helps in analyzing motion from different perspectives, which is crucial in physics.