Analysis:

When the ball is released at an angle of 45° with the vertical, it will experience a combination of gravitational force and the tension in the string. To determine the acceleration of the box at this instant, we need to consider the forces acting on the ball.

Forces Acting on the Ball:

1. Gravitational Force:
- The ball experiences a gravitational force acting vertically downwards. This force can be represented as mg, where m is the mass of the ball and g is the acceleration due to gravity.

2. Tension in the String:
- The string connecting the ball to the box exerts a tension force on the ball. This tension force acts along the string, towards the box.

Components of Forces:

The gravitational force and the tension force can be resolved into their respective components. Since the ball is released at an angle of 45° with the vertical, the components of these forces can be calculated as follows:

1. Gravitational Force Components:
- The vertical component of the gravitational force is mg * cos(45°).
- The horizontal component is mg * sin(45°).

2. Tension Force Components:
- The vertical component of the tension force is T * cos(45°), where T is the tension in the string.
- The horizontal component is T * sin(45°).

Net Force:

To determine the acceleration of the box, we need to find the net force acting on the ball. The net force can be calculated by summing up the vertical and horizontal components of the forces.

1. Vertical Forces:
- The vertical component of the gravitational force is mg * cos(45°).
- The vertical component of the tension force is T * cos(45°).
- The net vertical force is the sum of these two components: mg * cos(45°) + T * cos(45°).

2. Horizontal Forces:
- The horizontal component of the gravitational force is mg * sin(45°).
- The horizontal component of the tension force is T * sin(45°).
- The net horizontal force is the sum of these two components: mg * sin(45°) + T * sin(45°).

Acceleration:

The acceleration of the box can be calculated using Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.

1. Vertical Acceleration:
- The net vertical force is mg * cos(45°) + T * cos(45°).
- The mass of the ball is equal to the mass of the box (let's assume it is m).
- Therefore, the vertical acceleration is (mg * cos(45°) + T * cos(45°)) / m.

2. Horizontal Acceleration:
- The net horizontal force is mg * sin(45°) + T * sin(45°).
- Since the tension force is acting towards the box, the horizontal acceleration will be in the opposite direction.
- Therefore, the horizontal acceleration is -(mg * sin(45°) + T * sin(45°)) / m.

Conclusion:

To determine the acceleration of the box when the ball is released at an angle of 45° with the vertical, we need to calculate

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