A spring with a constant of #4 (kg)/s^2# is lying on the ground with one end attached to a wall. An object with a mass of #6 kg# and speed of #9 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

This question relates to energy conservation, as you have already observed.

In the spring, all of the object's kinetic energy will transform into elastic potential energy.

This indicates that the amount of force needed to compress or stretch the spring depends on how much of it has already been done so.

In terms of math, we write,

The force will always oppose the spring's displacement because the spring always seeks to return to its equilibrium length, which is the source of the negative sign.

To find the compression of the spring, we can use the principle of conservation of mechanical energy. Initially, the object has kinetic energy, which is converted into potential energy stored in the spring when it compresses. The equation for the conservation of mechanical energy is:

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

Where: m = mass of the object (6 kg) v = velocity of the object (9 m/s) k = spring constant (4 kg/s^2) x = compression of the spring (unknown)

Plugging in the values:

1/2 * 6 * (9)^2 = 1/2 * 4 * x^2

Solving for x:

(1/2) * 6 * 81 = (1/2) * 4 * x^2 243 = 2 * x^2 x^2 = 243 / 2 x^2 = 121.5 x = √121.5 x ≈ 11 meters

So, the spring will compress approximately 11 meters.