You set up a ramp on your desk at home. Marbles rolling down the ramp and horizontally off your desk consistantly land 52.0 cm from the base of your desk. Your desk is 69.0 cm high. If you pull your desk over to the window of your second story room and "launch" marbles to the ground (6.48 meters below the desk top), how far out into the yard will the marbles land?

I will be happy to critique your thinking.

To find out how far the marbles will land in the yard when launched from the window, we can use the principles of projectile motion. Here's how we can calculate it:

1. First, we need to find the horizontal velocity of the marbles when they leave the desk. Since the marbles roll horizontally off the desk, there is no vertical component to their velocity. We can assume that the horizontal velocity remains constant during the entire flight.

2. We can use the formula for horizontal displacement (range) in projectile motion:

Range = Horizontal Velocity × Time of flight

Since the marbles are initially at rest when they roll off the desk, the vertical displacement (height) of the desk will be equal to the vertical displacement of the marbles during their flight.

3. To find the time of flight, we need to calculate the time it takes for the marbles to fall from the desk to the ground.

We can use the formula:

Distance = (1/2) × acceleration × time^2

Considering acceleration due to gravity as 9.8 m/s^2, and the distance as 6.48 meters, we can rearrange the equation to solve for time.

4. Once we have the time of flight, we can calculate the horizontal displacement (range) using the previously found horizontal velocity formula.

Now let's perform the calculations:

1. Calculating the time of flight:
Distance = (1/2) × acceleration × time^2
6.48 m = (1/2) × 9.8 m/s^2 × time^2

Rearranging the equation to solve for time^2:
time^2 = (2 × distance) / acceleration
time^2 = (2 × 6.48 m) / 9.8 m/s^2
time^2 = 1.32 s
time = √1.32 s (taking the square root)

Therefore, the time of flight is approximately 1.15 seconds.

2. Calculating the horizontal velocity:
To calculate the horizontal velocity, we can use the horizontal distance the marbles travelled on the desk.

Horizontal distance = 52.0 cm = 0.52 m
Velocity = distance / time
Velocity = 0.52 m / 1.15 s
Velocity ≈ 0.45 m/s

3. Calculating the range (horizontal displacement):
Range = Velocity × Time of flight
Range = 0.45 m/s × 1.15 s
Range ≈ 0.52 m

Therefore, the marbles will land approximately 0.52 meters out into the yard when launched from the second-story window.