A ball is attached to one end of a wire, the other end being fastened to the ceiling. The wire is held horizontal, and the ball is released from rest (see the drawing). It swings downward and strikes a block initially at rest on a horizontal frictionless surface. Air resistance is negligible, and the collision is elastic. The masses of the ball and block are, respectively, 1.8 kg and 2.5 kg, and the length of the wire is 1.11 m. Find the velocity (magnitude and direction) of the ball (a) just before the collision, and (b) just after the collision.

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To find the velocity of the ball just before the collision, we can use the principle of conservation of mechanical energy. Before the collision, the ball swings downward from rest, which means its initial potential energy is converted into kinetic energy.

To calculate the velocity just before the collision, we need to find the potential energy at the highest point of the swing. The ball reaches its highest point when the wire is vertical, so the potential energy at that point is given by:

Potential energy = mass * gravity * height

Height = Length of the wire - Length of the wire * cos(angle)

The angle can be found using the length of the wire and the vertical displacement of the ball. Since the wire is held horizontal, the angle formed by the wire and the vertical line is 90 degrees. Therefore, the angle can be calculated using the equation:

Angle = arcsin(vertical displacement / length of the wire)

Now we can calculate the potential energy at the highest point:

Potential energy = mass * gravity * (Length of the wire - Length of the wire * cos(angle))

Next, we equate the potential energy at the highest point to the kinetic energy just before the collision:

Potential energy = Kinetic energy

m * g * (Length of the wire - Length of the wire * cos(angle)) = 0.5 * m * v^2

Simplifying the equation, we get:

g * (Length of the wire - Length of the wire * cos(angle)) = 0.5 * v^2

Plugging in the known values, we can solve for v:

v^2 = 2 * g * (Length of the wire - Length of the wire * cos(angle)) / m

Once we have the magnitude of the velocity just before the collision, we can find its direction by considering the direction of motion.

Now, to find the velocity of the ball just after the collision, we need to consider that the collision is elastic. In an elastic collision, both kinetic energy and momentum are conserved.

We can calculate the momentum before the collision using the mass and velocity just before the collision:

Momentum before collision = mass * velocity (just before the collision)

To find the momentum just after the collision, we need to consider that the ball and the block are connected, and after the collision, they move together.

Therefore, the total mass after the collision is the combined mass of the ball and the block.

Momentum after collision = (mass of the ball + mass of the block) * velocity (just after the collision)

Since momentum is conserved, we have:

Momentum before collision = Momentum after collision

mass * velocity (just before the collision) = (mass of the ball + mass of the block) * velocity (just after the collision)

Simplifying the equation, we get:

velocity (just after the collision) = (mass * velocity (just before the collision)) / (mass of the ball + mass of the block)

Plugging in the known values, we can find the magnitude and direction of the velocity just after the collision.

Note: Remember to convert the angle to radians when using trigonometric functions.