Final answer:
To hit the target box, Rhoda needs to compress the spring more than Bobby, who compressed it by 1.10 cm but fell 27.0 cm short of the 2.20 m distance. The required compression can be found using the ratio of the distances travelled by Bobby's marble and the desired distance.
Step-by-step explanation:
When two children are playing a game where they try to hit a small box with a marble fired from a spring-loaded gun mounted on a table, we are looking at a problem of projectile motion and energy. The required compression of the spring correlates with the kinetic energy transferred to the marble, which in turn determines the horizontal distance it travels. Since friction is ignored, we assume that all energy stored in the spring gets transferred to the marble.
Bobby's attempt falls 27.0 cm short of the horizontal distance d, which implies that the initial kinetic energy and therefore the compression of the spring was insufficient. To solve for Rhoda's necessary compression, we assume that the relationship between the compression distance and the distance travelled by the marble is linear since spring force, in this case, is directly proportional to the displacement from its equilibrium position (Hooke's Law).
Given Bobby's compression of the spring was 1.10 cm, Rhoda would need to compress the spring by a distance x such that:
- The ratio of Rhoda's compression to Bobby's compression should be the same as the ratio of the desired travel distance to Bobby's travel distance.
- Therefore, x / 1.10 cm = 2.20 m / (2.20 m - 0.27 m).
- Solving for x gives us the needed compression for Rhoda to make a direct hit.