### Analysis of Curvilinear Motion of a Connecting PinThe x- and y-motions of guides **A** and **B** with right-angle slots control the curvilinear motion of the connecting pin **P**, which slides in both slots. For a short interval, the motions are governed by:\[ x = 12 + 0.55t^2 \]\[ y = 16 - 0.57t^3 \]where \( x \) and \( y \) are in millimeters and \( t \) is in seconds. Calculate the magnitudes of the velocity \( v \) and acceleration \( a \) of the pin for \( t = 2.5 \, s \). Sketch the direction of the path and indicate its curvature for this instant.Below is a diagram:![Diagram](diagram_url)**Explanation of the Diagram:**- The diagram illustrates a mechanical setup where the guides **A** and **B** move in perpendicular directions.- **A** moves horizontally along the x-axis, and **B** moves vertically along the y-axis, controlling the position of the pin **P**.- The springs attached to the guides exert forces to control the motion of **P**.### Part 1#### Calculate the x- and y-components of the velocity.The velocity components can be found by differentiating the position functions with respect to time.1. For \( x = 12 + 0.55t^2 \):\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(12 + 0.55t^2) = 1.10t \]2. For \( y = 16 - 0.57t^3 \):\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(16 - 0.57t^3) = -1.71t^2 \]For \( t = 2.5 \, s \):\[ v_x = 1.10 \times 2.5 = 2.75 \, \text{mm/s} \]\[ v_y = -1.71 \times (2.5)^2 = -10.6875 \, \text{mm/s} \]#### Answers\[ v_x = \boxed{2.75} \, \text{mm/s} \]\[ v_y =

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The x-and y-motions of guides A and B with right-angle slots control the curvilinear motion of the connecting pin P, which slides in both slots. For a short interval, the motions are governed by x= 12 +0.55t2 and y=16-0.57t³, where x and y are in millimeters and tis in seconds. Calculate the magnitudes of the velocity v and acceleration a of the pin for t = 2.5 s. Sketch the direction of the path and indicate its curvature for this instant. B Part 1 Calculate the x- and y-components of the velocity. Answers: Vx² mm/s Vy= mm/s i i

The x-and y-motions of guides A and B with right-angle slots control the curvilinear motion of the connecting pin P, which slides in both slots. For a short interval, the motions are governed by x= 12 +0.55t2 and y=16-0.57t³, where x and y are in millimeters and tis in seconds. Calculate the magnitudes of the velocity v and acceleration a of the pin for t = 2.5 s. Sketch the direction of the path and indicate its curvature for this instant. B Part 1 Calculate the x- and y-components of the velocity. Answers: Vx² mm/s Vy= mm/s i i

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Question

### Analysis of Curvilinear Motion of a Connecting Pin

The x- and y-motions of guides **A** and **B** with right-angle slots control the curvilinear motion of the connecting pin **P**, which slides in both slots. For a short interval, the motions are governed by:
\[ x = 12 + 0.55t^2 \]
\[ y = 16 - 0.57t^3 \]
where \( x \) and \( y \) are in millimeters and \( t \) is in seconds. Calculate the magnitudes of the velocity \( v \) and acceleration \( a \) of the pin for \( t = 2.5 \, s \). Sketch the direction of the path and indicate its curvature for this instant.

Below is a diagram:
![Diagram](diagram_url)

**Explanation of the Diagram:**
- The diagram illustrates a mechanical setup where the guides **A** and **B** move in perpendicular directions.
- **A** moves horizontally along the x-axis, and **B** moves vertically along the y-axis, controlling the position of the pin **P**.
- The springs attached to the guides exert forces to control the motion of **P**.

### Part 1

#### Calculate the x- and y-components of the velocity.

The velocity components can be found by differentiating the position functions with respect to time.
1. For \( x = 12 + 0.55t^2 \):
\[ v_x = \frac{dx}{dt} = \frac{d}{dt}(12 + 0.55t^2) = 1.10t \]

2. For \( y = 16 - 0.57t^3 \):
\[ v_y = \frac{dy}{dt} = \frac{d}{dt}(16 - 0.57t^3) = -1.71t^2 \]

For \( t = 2.5 \, s \):
\[ v_x = 1.10 \times 2.5 = 2.75 \, \text{mm/s} \]
\[ v_y = -1.71 \times (2.5)^2 = -10.6875 \, \text{mm/s} \]

#### Answers
\[ v_x = \boxed{2.75} \, \text{mm/s} \]
\[ v_y =

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