in a pinball machine, a force of 24 N is used to compress a spring a distance of 0.080m.

a. What is the potential energy stored in the spring?
b. What is the maximum speed tha it van produve in a ball of mass 0.15kg?

k=F/x=24/0.08 = 300 N/m

PE = kx²/2 = 300•(0.08)²/2= 0.96 J.

KE=PE
mv²/2 =kx²/2
v=sqrt(kx²/m) = sqrt(300•(0.08)²/0.15) = 3.58 m/s

To calculate the potential energy stored in the spring, you can use the formula:

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

Where:
PE is the potential energy stored in the spring
k is the spring constant
x is the compression distance

Since the force is used to compress the spring, we can find the spring constant (k) using Hooke's Law:

Force (F) = k * x

Rearranging the equation, we get:

k = F / x

a. To find the potential energy stored in the spring:

Step 1: Calculate the spring constant (k)
k = F / x
k = 24 N / 0.080 m
k = 300 N/m

Step 2: Calculate the potential energy stored in the spring
PE = (1/2) * k * x^2
PE = (1/2) * 300 N/m * (0.080 m)^2
PE = 0.96 J

Therefore, the potential energy stored in the spring is 0.96 Joules.

b. To find the maximum speed that it can produce in a ball of mass 0.15 kg:

Step 1: Use the potential energy formula (PE = (1/2) * m * v^2), where m is the mass and v is the velocity.

Potential Energy (PE) = (1/2) * m * v^2

Step 2: Rearrange the equation to solve for velocity (v):

v^2 = (2 * PE) / m

Step 3: Substitute the values:
PE = 0.96 J
m = 0.15 kg

v^2 = (2 * 0.96 J) / 0.15 kg

v^2 ≈ 12.8 m^2/s^2

Step 4: Take the square root of both sides to find the velocity:

v ≈ √(12.8 m^2/s^2)
v ≈ 3.58 m/s

Therefore, the maximum speed that it can produce in a ball of mass 0.15 kg is approximately 3.58 m/s.

To find the potential energy stored in the spring, you can use the formula for the potential energy of a spring:

Elastic potential energy (PE) = (1/2) k x^2

Where:
- PE is the potential energy stored in the spring
- k is the spring constant (a measure of the stiffness of the spring)
- x is the displacement of the spring from its equilibrium position

In this case, the force used to compress the spring is given as 24 N and the displacement is given as 0.080 m.

Step-by-step solution for part a:
1. We need to first find the spring constant (k). The spring constant can be calculated using Hooke's Law equation: F = kx, where F is the force and x is the displacement.
k = F / x = 24 N / 0.080 m = 300 N/m

2. Substitute the spring constant and displacement into the formula for potential energy:
PE = (1/2) k x^2
PE = (1/2) * 300 N/m * (0.080 m)^2

Calculate the potential energy to get the answer.

To find the maximum speed that can be produced in a ball of mass 0.15 kg, you can use the principle of conservation of mechanical energy. The potential energy stored in the spring will be converted to the kinetic energy of the ball.

Step-by-step solution for part b:
1. The potential energy stored in the spring (PE) is given by the equation PE = (1/2) k x^2. Calculate the potential energy using the spring constant (k) and displacement (x) obtained in part a.

2. The potential energy will be converted into kinetic energy (KE) when the ball is released. The equation for kinetic energy is KE = (1/2) m v^2, where m is the mass of the ball and v is its velocity.

3. Equate the potential energy (PE) to the kinetic energy (KE):
PE = KE
(1/2) k x^2 = (1/2) m v^2

4. Rearrange the equation to find the velocity (v):
v = √((k x^2) / m)

5. Substitute the values into the equation and calculate the maximum speed.

Note: In this case, we assume that all the potential energy is converted into kinetic energy, neglecting any energy losses due to factors such as friction.