Two particles A and B r confined to always be in a circular groove of radius √17 m. At the same time , the particles must also be in a slot that has the shape of a parabola , which has an eqn. x = y² at t=0. The slot is shown dashed at time t=0. If the slot moves to the right at a constant speed of 3 m/s , find the speed of A at t= 1 sec.

To determine the speed of particle A at t = 1 second, we need to analyze the motion of both particles A and B as they are constrained by 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, confined to a circular groove with a radius of √17 meters. Additionally, they must remain within a parabolic slot defined by the equation x = y² at t = 0. The slot moves to the right at a constant speed of 3 m/s.

Position of the Parabola Over Time

At t = 0, the vertex of the parabola is at the origin (0,0). As time progresses, the entire parabola shifts to the right. The position of the parabola at any time t can be expressed as:

• x = y² + 3t

This means that at t = 1 second, the equation of the parabola becomes:

• x = y² + 3(1) = y² + 3

Finding the Circular Groove's Equation

The circular groove has a radius of √17, which means the equation of the circle can be expressed as:

• x² + y² = 17

Finding the Intersection Points

To find the points where the circular groove intersects with the moving parabolic slot at t = 1 second, we need to solve the two equations simultaneously:

• x² + y² = 17
• x = y² + 3

Substituting the second equation into the first gives:

• (y² + 3)² + y² = 17

Expanding this, we have:

• y⁴ + 6y² + 9 + y² = 17
• y⁴ + 7y² - 8 = 0

This is a quadratic in terms of y². Letting z = y², we can rewrite it as:

• z² + 7z - 8 = 0

Solving the Quadratic Equation

Using the quadratic formula, z = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = 7, and c = -8:

• z = (-7 ± √(49 + 32)) / 2
• z = (-7 ± √81) / 2
• z = (-7 ± 9) / 2

This gives us two solutions:

• z = 1 (y² = 1, y = ±1)
• z = -8 (not valid since z = y² cannot be negative)

Thus, we have two points of intersection: (4, 1) and (4, -1) when substituting y = ±1 back into x = y² + 3.

Calculating the Speed of Particle A

Now that we have the position of particle A at t = 1 second, we can find its speed. The speed of a particle in circular motion can be calculated using the formula:

• v = rω

Where r is the radius and ω is the angular velocity. Since the particles are constrained to the circular groove, we need to find the angular velocity based on the position of the particles.

Angular Velocity Calculation

At the intersection points, the radius remains constant at √17 m. The angular velocity can be derived from the linear speed of the slot (3 m/s) and the radius:

• ω = v/r = 3/√17

Thus, the speed of particle A at t = 1 second is:

• v = √17 * (3/√17) = 3 m/s

Final Result

Therefore, the speed of particle A at t = 1 second is 3 m/s. This speed reflects the constant motion of the parabolic slot and the constraints of the circular groove.