Problem:
A block of mass m moving with speed v compresses a spring through distance x before its speed is halved. What is the value of the spring constant?

Solution:

To find the value of the spring constant, we can use the principle of conservation of mechanical energy. The initial kinetic energy of the block is equal to the potential energy stored in the compressed spring.

Key Points:
- Use the principle of conservation of mechanical energy to find the spring constant.
- The initial kinetic energy of the block is equal to the potential energy stored in the spring.
- The potential energy stored in the spring is given by the equation U = (1/2)kx^2.
- The initial kinetic energy of the block is given by the equation KE = (1/2)mv^2.

Step 1: Identify the initial and final states:
- Initial state: The block is moving with speed v.
- Final state: The block's speed is halved, which means it is moving with speed v/2.

Step 2: Calculate the initial kinetic energy:
The initial kinetic energy of the block is given by the equation KE = (1/2)mv^2.

Step 3: Calculate the potential energy stored in the spring:
The potential energy stored in the spring is given by the equation U = (1/2)kx^2, where k is the spring constant and x is the distance the spring is compressed.

Step 4: Equate the initial kinetic energy to the potential energy:
Setting the initial kinetic energy equal to the potential energy, we have:
(1/2)mv^2 = (1/2)kx^2

Step 5: Solve for the spring constant:
To find the value of the spring constant, we rearrange the equation:
k = (mv^2)/x^2

Step 6: Substitute the given values:
Substitute the given values of mass m, speed v, and distance x into the equation:
k = (m(v^2))/(x^2)

Step 7: Simplify the equation:
Simplify the equation by canceling out common terms:
k = (m(v^2))/(x^2)

Conclusion:
The value of the spring constant is given by the equation k = (m(v^2))/(x^2). Use the given values of mass m, speed v, and distance x to calculate the spring constant.

3mv2÷4x2