A mass of 1.8 kg traveling at 6.9 m/s along a smooth, horizontal plane hits a relaxed spring. (Assume no friction at the interface between the mass and the plane).The mass is slowed to zero velocity when the spring has been compressed by 0.43 m.What is the spring constant of the spring?Express your answer in N/m, to at least one digit after the decimal point.

To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

First, let's calculate the change in potential energy of the mass-spring system. The potential energy stored in a spring can be calculated using the formula:

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

where k is the spring constant and x is the displacement of the spring.

In this case, the mass is slowed to zero velocity, so all of its initial kinetic energy is converted into potential energy stored in the spring. Therefore, we can equate the change in potential energy to the initial kinetic energy of the mass:

(1/2) * k * x^2 = (1/2) * m * v^2

where m is the mass of the object and v is its initial velocity.

Plugging in the given values: m = 1.8 kg v = 6.9 m/s x = 0.43 m

(1/2) * k * (0.43)^2 = (1/2) * 1.8 * (6.9)^2

Simplifying the equation: k * (0.43)^2 = 1.8 * (6.9)^2

Now, solve for k: k = (1.8 * (6.9)^2) / (0.43)^2

Calculating the value: k ≈ 207.6 N/m

Therefore, the spring constant of the spring is approximately 207.6 N/m.

To find the spring constant of the spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position.

First, let's calculate the change in potential energy of the mass-spring system. The potential energy stored in a spring can be calculated using the formula:

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

where k is the spring constant and x is the displacement of the spring.

In this case, the mass is slowed to zero velocity, so all of its initial kinetic energy is converted into potential energy stored in the spring. Therefore, we can equate the change in potential energy to the initial kinetic energy of the mass:

(1/2) * k * x^2 = (1/2) * m * v^2

where m is the mass of the object and v is its initial velocity.

Plugging in the given values: m = 1.8 kg v = 6.9 m/s x = 0.43 m

(1/2) * k * (0.43)^2 = (1/2) * 1.8 * (6.9)^2

Simplifying the equation: k * (0.43)^2 = 1.8 * (6.9)^2

Now, we can solve for the spring constant k:

k = (1.8 * (6.9)^2) / (0.43)^2

Calculating the value: k ≈ 207.6 N/m

Therefore, the spring constant of the spring is approximately 207.6 N/m.