A 2.9-kg block is released from rest and allowed to slide down a frictionless surface and into a spring. The far end of the spring is attached to a wall, as shown. The initial height of the block is 0.34 m above the lowest part of the slide and the spring constant is 347 N/m.

(a) What is the block's speed when it is at a height of 0.25 m above the base of the slide?

(b) How far is the spring compressed?

(c) The spring sends the block back to the left. How high does the block rise?

To solve this problem, we can use the principles of conservation of mechanical energy and Hooke's Law.

(a) To find the block's speed when it is at a height of 0.25 m above the base of the slide, we can use the principle of conservation of mechanical energy.

The initial mechanical energy of the block is given by the potential energy at the initial height:
Ep = m * g * h1

The final mechanical energy of the block is given by the sum of its potential energy at the final height and its kinetic energy.
Ef = m * g * h2 + (1/2) * m * v^2

Since there is no friction, the total mechanical energy remains constant. Therefore, we can equate Ep and Ef:

m * g * h1 = m * g * h2 + (1/2) * m * v^2

Simplifying the equation, we get:

0.34 * 9.8 = 0.25 * 9.8 + (1/2) * v^2

Solving for v (the block's speed), we can rearrange the equation as:

v^2 = 2 * (0.34 - 0.25) * 9.8

v^2 = 2 * 0.09 * 9.8

v^2 = 17.64

v ≈ 4.2 m/s

Therefore, the block's speed when it is at a height of 0.25 m above the base of the slide is approximately 4.2 m/s.

(b) To find how far the spring is compressed, we can use Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

F = -k * x

Where F is the force, k is the spring constant, and x is the displacement.

At the equilibrium position, the force exerted by the spring is zero. Therefore, we can equate the weight of the block to the force exerted by the compressed spring:

m * g = k * x

Solving for x (the compression of the spring), we get:

x = (m * g) / k

x = (2.9 * 9.8) / 347

x ≈ 0.082 m

Therefore, the spring is compressed approximately 0.082 m.

(c) To find how high the block rises when the spring sends it back to the left, we can again use the principle of conservation of mechanical energy.

The initial mechanical energy of the block is given by the sum of its potential energy at the final position and its kinetic energy:
Ei = m * g * h2 + (1/2) * m * v^2

The final mechanical energy of the block is given by its potential energy at the maximum height:
Ef = m * g * h3

Since there is no friction, the total mechanical energy remains constant. Therefore, we can equate Ei and Ef:

m * g * h2 + (1/2) * m * v^2 = m * g * h3

Simplifying the equation, we get:

h3 = h2 + (v^2) / (2 * g)

Plugging in the values:

h3 ≈ 0.25 + (4.2^2) / (2 * 9.8)

h3 ≈ 0.477 m

Therefore, the block rises approximately 0.477 m.