Marbles rolling down the ramp and horizontally off your desk consistantly land 72.0 cm from the base of your desk. Your desk is 77.0 cm high. If you pull your desk over to the window of your second story room and "launch" marbles to the ground (6.28 meters below the desk top), how far out into the yard will the marbles land?
To find out how far the marbles will land in the yard, we can use the principle of conservation of energy.
Let's analyze the situation step by step:
1. Determine the vertical distance traveled by the marble from the desk to the ground:
The height of the desk is given as 77.0 cm, and the ground is 6.28 meters below the desk. In order to find the vertical distance traveled, we need to subtract the height of the desk from the total distance to the ground:
Total vertical distance = 77.0 cm + 6.28 m = 77.0 cm + 628.0 cm = 705.0 cm.
2. Calculate the horizontal distance traveled by the marble based on the vertical distance:
In the initial scenario, where the marbles consistently land 72.0 cm from the base of the desk, we can assume that there is no horizontal force acting on the marbles. Therefore, the horizontal distance traveled is independent of the vertical distance.
From the given information, we know that the marbles consistently land 72.0 cm (or 0.72 meters) from the base of the desk.
3. Calculate the equivalent horizontal distance from the desk to the yard:
We can use similar triangles to determine the equivalent horizontal distance from the desk to the yard. Since the height from the desk to the ground is 705.0 cm (or 7.05 meters), and the horizontal distance from the desk to the ground is 0.72 meters, we can set up the following proportion:
height of yard / horizontal distance of yard = height of desk / horizontal distance of desk.
Plugging in the values, we get:
height of yard / horizontal distance of yard = 7.05 m / 0.72 m.
Rearranging the equation to solve for horizontal distance of yard, we get:
horizontal distance of yard = (height of yard * horizontal distance of desk) / height of desk.
Plugging in the values, we get:
horizontal distance of yard = (7.05 m * 0.72 m) / 77.0 cm.
Converting the height of desk to meters, we get:
horizontal distance of yard = 5.076 m.
Therefore, the marbles will land approximately 5.076 meters out into the yard.