Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring-loaded gun that is mounted on a table.

Question

Two children are playing a game in which they try to hit a small box on the floor with a marble fired from a spring-loaded gun that is mounted on a table.

(Image included with problem)
The target box is 2.44 m horizontally from the edge of the table. Bobby compresses the spring 1.42 cm, but the center of the marble falls 84.0 cm short of the center of the box. How far should Rhoda compress the spring to score a direct hit?

Answer 1

You want the muzzle velocity to be more by a factor of (2.44/(2.44-.84)

KE=springPE
1/2 m v^2=1/2 kx^2
so if v increases by above,
then x^2 increases by (2.44/(2.44-.84))^2

new cmpression=1.42cm*(2.44/(2.44-.84))^2

Answer 2

To find how far Rhoda should compress the spring to score a direct hit, we need to first determine the horizontal distance the marble travels when Bobby compresses the spring by 1.42 cm.

Let's calculate the horizontal distance the marble travels with Bobby's spring compression:

Given:
Compressed spring = 1.42 cm (or 0.0142 m)
Horizontal distance from table edge to the target box = 2.44 m

We know that the horizontal distance traveled by the marble is directly proportional to the spring compression. Therefore, we can set up a proportion using similar triangles:

(horizontal distance with Bobby's compression) / (horizontal distance with Rhoda's compression) = (spring compression distance with Bobby's compression) / (spring compression distance with Rhoda's compression)

Let's denote:
D_b = horizontal distance with Bobby's compression
D_r = horizontal distance with Rhoda's compression

The proportion can be written as:
D_b / D_r = (0.0142 m) / (x m)
where x is the spring compression distance with Rhoda's compression.

Now, we can solve for D_b, the horizontal distance with Bobby's compression.

We are given that the center of the marble falls 84.0 cm short of the center of the box, which means it stops 84.0 cm before reaching the box.

Therefore, D_b = (2.44 m) - (0.84 m) = 1.60 m
(We subtract 0.84 m because the marble falls 84.0 cm short of the box)

Now, we can plug in the known values into the proportion and solve for x:

1.60 m / D_r = 0.0142 m / x

To solve for x, we can rearrange the equation:

x = (0.0142 m) * D_r / 1.60 m

Now that we have the equation to find x, we can solve for it.

Please provide the value for D_r, the horizontal distance with Rhoda's compression, in order to calculate the required spring compression for a direct hit.