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.
The target box is 3.46 m horizontally from the edge of the table. Bobby compresses the spring 0.97 cm, but the center of the marble falls 73.0 cm short of the center of the box. How far should Rhoda compress the spring to score a direct hit?
To solve this problem, we need to use the principles of projectile motion.
First, let's define some variables:
- D1 = distance that Bobby compresses the spring (0.97 cm)
- D2 = distance that Rhoda needs to compress the spring
- X = horizontal distance from the edge of the table to the target box (3.46 m)
- Y = vertical distance between the center of the marble and the center of the box (73.0 cm)
We'll start by finding the initial velocity of the marble when Bobby fired it. We can use the formula for the potential energy stored in a spring:
PE = 0.5 * k * D^2
where PE is the potential energy, k is the spring constant, and D is the distance the spring is compressed. Since the question doesn't provide the spring constant, we can ignore it for now and treat it as a constant that cancels out.
Next, we'll use the equation for horizontal projectile motion:
X = V0x * t
where X is the horizontal distance, V0x is the initial horizontal velocity, and t is the time of flight.
To find V0x, we need to calculate the time of flight using the equation for vertical projectile motion:
Y = V0y * t + 0.5 * g * t^2
where Y is the vertical distance, V0y is the initial vertical velocity, and g is the acceleration due to gravity.
Since the marble reaches its maximum height halfway through the flight, V0y at the highest point is 0. This means we can find the time of flight as follows:
t = sqrt(2Y/g)
Now that we have the time of flight, we can find the initial horizontal velocity:
V0x = X / t
Finally, we can use the equation for the potential energy of the spring again to find the distance Rhoda needs to compress the spring:
D2 = sqrt(2 * PE / spring constant)
However, since the spring constant cancels out, we can rewrite the equation as:
D2 = sqrt(2 * PE)
Now let's plug in the values and calculate the answer.