A block of mass m is initially moving to the right on a horizontal frictionless surface at a speed v. It then compresses a spring of spring constant k. At the instant when the kinetic energy
of the block is equal to the potential energy of the spring, the spring is compressed a distance of:

The distance that the spring compresses is:

Explanation:

Kinetic and Elastic Potential Energy

The kinetic energy of an object of mass m traveling at a speed v is:

The elastic potential energy of a spring of constant k that compresses a distance x is:

The block of mass m is moving at a speed v when compresses a spring of constant k. The kinetic energy will eventually transform into elastic energy, but before that, both energies will be equal. It happens when:

Simplifying:

Dividing by k:

Taking square root:

The distance that the spring compresses is

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west