Aspring is resting vertically on a table. a small box is dropped onto the top of the spring and compresses it. suppose the spring has a spring constant of 450 n/m and the box has a mass of 1.5 kg. the speed of the box just before it makes contact with the spring is 0.49 m/s.(a) determine the magnitude of the spring's displacement at an instant when the acceleration of the box is zero.(b) what is the magnitude of the spring's displacement when the spring is fully compressed? (cutnell 285) cutnell, john d. physics, ap edition, 11th edition. wiley, 2018-06-27. vitalbook file. the citation provided is a guideline. check each citation for accuracy before use.

(a) 0.033 m

There are two forces acting on the box as it touches the spring:

- Gravity: , downward, with m=1.5 kg being the mass of the box and g=9.8 m/s^2 being the gravitational acceleration

- The force of the spring: , upward, with k=450 N/m being the spring constant x being the displacement of the spring

According to Newton's second law, we have:

when the acceleration of the box is zero, a=0, so the previous equation becomes

And solving for x we find the displacement of the spring when the box's acceleration is zero:

(b) 0.028 m

When the spring is fully compressed, the box's speed is zero, so this means that the whole initial kinetic energy of the box has been converted into elastic potential energy of the spring:

where v=0.49 m/s is the initial speed of the spring. Solving for x, we find the compression of the spring: