To find the speed of particle A at t = 1 second, we need to analyze the motion of both particles A and B within the constraints of the circular groove and the parabolic slot. Let's break down the problem step by step.

Understanding the Setup

We have two particles, A and B, constrained to move in a circular groove with a radius of √17 meters. Additionally, they are confined to a parabolic slot described by the equation x = y². At t = 0, the slot is positioned such that it intersects the circular groove.

Movement of the Slot

The slot moves to the right at a constant speed of 3 m/s. This means that after 1 second, the position of the slot will have shifted to the right by:

• Distance = Speed × Time = 3 m/s × 1 s = 3 m

Thus, at t = 1 second, the equation of the slot will be:

x = y² - 3

Finding the Position of Particle A

At t = 1 second, we need to determine the coordinates of particle A. Since it is constrained to the circular groove, we can use the equation of the circle:

x² + y² = (√17)² = 17

We also need to satisfy the new equation of the slot, which is x = y² - 3. Substituting this into the circle's equation gives:

(y² - 3)² + y² = 17

Solving for y

Expanding the equation:

• (y² - 3)² = y⁴ - 6y² + 9
• Thus, the equation becomes: y⁴ - 6y² + 9 + y² = 17
• Which simplifies to: y⁴ - 5y² - 8 = 0

Using Substitution

Let z = y². The equation becomes:

z² - 5z - 8 = 0

We can solve this quadratic equation using the quadratic formula:

z = [5 ± √(25 + 32)] / 2 = [5 ± √57] / 2

Calculating the roots gives us two possible values for z, and consequently for y:

• y = ±√[(5 + √57)/2]
• y = ±√[(5 - √57)/2]

Calculating the Speed of Particle A

To find the speed of particle A, we need to determine its velocity components. The velocity in the circular groove can be derived from the angular velocity, which is constant due to the circular motion. The speed can be calculated using the relationship:

v = rω, where r is the radius and ω is the angular velocity.

Finding Angular Velocity

Since the groove is circular, the angular velocity can be related to the linear speed of the particles. The total distance traveled by A in the circular path can be expressed in terms of the angle θ:

θ = ωt, where θ is in radians.

At t = 1 second, we can find the angle θ that corresponds to the position of A on the circle. The speed of A will depend on its position along the circular path and the movement of the slot.

Final Calculation

After determining the coordinates of A and the corresponding angle, we can compute the speed of A. The final speed will be a combination of the circular motion and the translational motion due to the slot's movement. The total speed of particle A at t = 1 second will be:

Speed of A = Speed due to circular motion + Speed due to slot movement

In summary, by analyzing the constraints of the circular groove and the moving parabolic slot, we can derive the speed of particle A at any given time, including t = 1 second. This approach combines geometry, algebra, and kinematics to arrive at a comprehensive solution.