A 2.50 mass is pushed against a horizontal spring of force constant 26.0 on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 15.0 of potential energy in it, the mass is suddenly released from rest.

a. Find the greatest speed the mass reaches?
b. What is the greatest acceleration of the mass?

The potential energy of the block equals the kinetic energy of the block, so:

1/2kx^2=1/2mv^2

Because it gives you the potential energy already, we can find the velocity.

15.0J=1/2(2.50kg)v^2
v=3.46m/s

I can't help you with the second part, though. Sorry!

a. Well, if I were the mass, I would definitely want to reach the greatest speed possible! It's the perfect opportunity for a thrilling roller coaster ride. Now, let's see. To find the greatest speed, we can make use of the law of conservation of energy. The potential energy stored in the spring gets converted into kinetic energy as the mass moves. So, we can equate the potential energy to the kinetic energy. 15.0 J of potential energy is equal to 1/2 * m * v^2, where v is the velocity. Now, just solve for v! Let the race for the greatest speed begin!

b. It's time for the mass to show off its acrobatic skills with some seriously impressive acceleration. To find the greatest acceleration, we can make use of good old Newton's second law - F = ma. In this case, the force acting on the mass is the force exerted by the compressed spring, which is given by Hooke's Law, F = -kx. Here, k is the force constant of the spring and x is the displacement of the spring from its equilibrium position. The negative sign just means that the force is acting in the opposite direction to the displacement. Now, divide both sides of the equation by the mass to get the acceleration, and you've got the answer! Feel the force, my friend!

To solve this problem, we can use the principle of conservation of energy. The potential energy stored in the spring is converted into kinetic energy as the mass is released. We can use the formula for potential energy in a spring and the formula for kinetic energy to find the answers.

a. To find the greatest speed the mass reaches, we can equate the potential energy stored in the spring to the kinetic energy of the mass:

Potential Energy (PE) = Kinetic Energy (KE)

The potential energy stored in the spring is given as 15.0 J, and the kinetic energy can be calculated using the formula:

KE = (1/2)mv^2

where m is the mass and v is the velocity.

Substituting the given values:

15.0 J = (1/2)(2.50 kg)(v^2)

Simplifying the equation, we get:

15.0 J = 1.25 kg(v^2)

Dividing both sides by 1.25 kg:

v^2 = 12.0 m^2/s^2

Taking the square root of both sides:

v = √(12.0 m^2/s^2)

Therefore, the greatest speed the mass reaches is approximately 3.46 m/s.

b. The acceleration of the mass can be calculated using the equation:

KE = (1/2)mv^2 = (1/2)ma^2

Substituting the given values:

(1/2)(2.50 kg)(3.46 m/s)^2 = (1/2)(2.50 kg)(a^2)

Simplifying the equation, we get:

6.73 m^2/s^2 = 1.25 kg(a^2)

Dividing both sides by 1.25 kg:

a^2 = 5.38 m^2/s^2

Taking the square root of both sides:

a = √(5.38 m^2/s^2)

Therefore, the greatest acceleration of the mass is approximately 2.32 m/s^2.

To find the greatest speed and acceleration of the mass, we can apply the principles of conservation of energy and Hooke's law.

Step 1: Calculate the unstrained length of the spring.
Given the force constant of the spring (k = 26.0 N/m) and the potential energy stored in the spring (PE = 15.0 J), we can use the formula for potential energy in a spring to find the unstrained length.

PE = (1/2)kx²
15.0 J = (1/2)(26.0 N/m)(x²)
Divide both sides by (1/2)(26.0 N/m) to solve for x².
30.0 J/N = x²
x = √30.0 ≈ 5.48 m

Step 2: Determine the maximum compression of the spring.
Since the mass is pushed against the spring and released from rest, the potential energy is entirely converted into kinetic energy.

PE = (1/2)kx² = (1/2)mv²
Substituting the given values:
15.0 J = (1/2)(26.0 N/m)(5.48 m)² = (1/2)(26.0 N/m)(30.05 m²)
Divide both sides by (1/2)(26.0 N/m) to solve for m.
m = (15.0 J) / [0.5(26.0 N/m)(30.05 m²)] ≈ 0.0198 kg

Step 3: Calculate the maximum speed of the mass.
Using the concept of conservation of energy, we can equate the potential energy stored in the spring to the kinetic energy attained by the mass.

PE = KE
(1/2)kx² = (1/2)mv²
Substituting the known values:
(1/2)(26.0 N/m)(5.48 m)² = (1/2)(0.0198 kg)v²
Solve for v:
v² = [(26.0 N/m)(5.48 m)²] / (0.0198 kg)
v = √[(26.0 N/m)(5.48 m)² / (0.0198 kg)]

Step 4: Calculate the maximum acceleration of the mass.
Using Hooke's law, we can find the maximum acceleration of the mass when the spring is fully compressed.

F = kx
ma = kx
a = (kx) / m
a = (26.0 N/m)(5.48 m) / 0.0198 kg

These calculations will provide the answers to part a and part b of the question.