A vertical spring (ignore its mass), whose spring constant is 980 N/m, is attached to a table and is compressed 0.160 m.

(a) What speed can it give to a 0.400 kg ball when released?
(b) How high above its original position (spring compressed) will the ball fly?

To solve this problem, we can use the principles of conservation of energy. The potential energy stored in the compressed spring is converted into kinetic energy of the ball when it is released.

(a) To find the speed of the ball, we need to calculate the potential energy stored in the compressed spring. The potential energy equation for a spring is given by:

Potential energy (PE) = 1/2 * k * x^2

where k is the spring constant and x is the displacement of the spring from its equilibrium position.

In this case, the spring constant (k) is 980 N/m and the displacement (x) is 0.160 m. Plugging in these values, we get:

PE = 1/2 * 980 N/m * (0.160 m)^2
= 1/2 * 980 N/m * 0.0256 m^2
= 12.8 J

The potential energy stored in the spring is 12.8 J.

When the ball is released, all the potential energy is converted into kinetic energy. The kinetic energy (KE) equation is given by:

Kinetic energy (KE) = 1/2 * m * v^2

where m is the mass of the ball and v is its velocity (speed).

In this case, the mass (m) of the ball is 0.400 kg. Substituting the values into the equation, we have:

12.8 J = 1/2 * 0.400 kg * v^2
v^2 = (2 * 12.8 J) / 0.400 kg
v^2 = 64 m^2/s^2
v = √(64 m^2/s^2) = 8 m/s

Therefore, the speed of the ball when released is 8 m/s.

(b) To find the height the ball will reach, we can use the conservation of mechanical energy. When the ball reaches its maximum height, all of its initial kinetic energy is converted into potential energy at its highest point.

The potential energy at the highest point can be calculated using the equation:

Potential energy (PE) = m * g * h

where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height above its original position.

In this case, the mass (m) of the ball is 0.400 kg and the acceleration due to gravity (g) is 9.8 m/s^2.

12.8 J = 0.400 kg * 9.8 m/s^2 * h
h = 12.8 J / (0.400 kg * 9.8 m/s^2)
h ≈ 3.27 m

Therefore, the ball will fly approximately 3.27 meters above its original position when released from the compressed spring.