Marbles rolling down the ramp and horizontally off your desk consistently land 46.0 cm from the base of your desk. Your desk is 78.5 cm high. If you pull your desk over to the window of your second story room and "launch" marbles to the ground (6.04 meters below the desk top), how far out into the yard will the marbles land?
To calculate how far the marbles will land in the yard, we can use the conservation of energy principle.
The potential energy of the marbles at the top of the ramp is equal to the kinetic energy of the marbles when they leave the desk.
Let's calculate the initial potential energy of the marbles when they are on the desk:
Potential energy = mass * gravity * height
The mass of the marbles doesn't affect their potential energy because they are not in free fall, so let's assume the mass is 1 kg. The acceleration due to gravity is approximately 9.8 m/s².
Potential energy = 1 kg * 9.8 m/s² * 78.5 cm
Now, let's calculate the kinetic energy of the marbles when they leave the desk:
Kinetic energy = (1/2) * mass * velocity²
The mass is still assumed to be 1 kg. We need to find the velocity at which the marbles leave the desk. The distance they consistently land from the base of the desk (46.0 cm) can help us find the velocity using the equation for horizontal motion:
Distance = velocity * time
Since the marbles fall vertically from the desk to the ground, the time it takes is the same as the time it would take for the marbles to travel 46.0 cm horizontally.
Let's calculate the time it takes for the marbles to travel horizontally:
Distance = velocity * time
46.0 cm = velocity * time
Converting cm to meters:
0.46 m = velocity * time
Now, let's calculate the velocity:
Velocity = 0.46 m / time
Now that we have the velocity, let's calculate the kinetic energy:
Kinetic energy = (1/2) * 1 kg * (velocity)²
Now that we have the initial potential energy and the final kinetic energy, we can equate them and solve for the velocity at which the marbles hit the ground:
Potential energy = Kinetic energy
1 kg * 9.8 m/s² * 78.5 cm = (1/2) * 1 kg * (velocity)²
Converting cm to meters:
0.785 m = (1/2) * (velocity)²
Rearranging the equation:
(velocity)² = 2 * 0.785 m
Taking the square root of both sides:
velocity = √(2 * 0.785 m)
Now that we have the velocity at which the marbles hit the ground, let's calculate the distance they will travel in the yard, which is 6.04 meters below the desk:
Distance = velocity * time
The time it takes for the marbles to reach the ground is the same as the time it would take for the marbles to travel 6.04 meters vertically. Let's calculate the time:
Distance = (1/2) * gravity * time²
6.04 m = (1/2) * 9.8 m/s² * time²
Simplifying the equation:
12.08 m = 4.9 m/s² * time²
Solving for time:
time² = 12.08 m / 4.9 m/s²
Taking the square root of both sides:
time = √(12.08 m / 4.9 m/s²)
Now, let's calculate the distance the marbles will travel in the yard:
Distance = velocity * time
Substituting the known values:
Distance = √(2 * 0.785 m) * √(12.08 m / 4.9 m/s²)
Distance = √(2 * 0.785 m * 12.08 m / 4.9 m/s²)
Distance = √(1.57 m² * 12.08 m / 4.9 m/s²)
Distance = √(19.10056 m³/s²)
Distance = 4.37 meters
Therefore, the marbles will land approximately 4.37 meters out into the yard.
To determine how far the marbles will land out into the yard, we need to use the principle of horizontal projectile motion.
First, let's calculate the time it takes for a marble to fall from the desk to the ground. We can use the equation for free fall:
d = (1/2)gt^2
where d is the distance (6.04 meters), g is the acceleration due to gravity (9.8 m/s^2), and t is the time.
Rearranging the equation to solve for t:
t^2 = (2d) / g
t = sqrt((2d) / g)
t = sqrt((2 * 6.04) / 9.8)
t ≈ 0.782 seconds
Now, since the marble is rolling off the desk horizontally, the horizontal distance traveled will be the product of the horizontal velocity and the time.
To determine the horizontal velocity, we can use the vertical distance and the height of the desk. We know that the marble consistently lands 46.0 cm from the base of the desk, which is 78.5 cm high.
The horizontal distance is given by:
horizontal distance = vertical distance / tan(θ)
where θ is the angle of elevation. In this case, the angle of elevation can be calculated using the height of the desk and the horizontal distance landed:
θ = arctan(46.0 cm / 78.5 cm)
Once we have the angle of elevation, we can calculate the horizontal velocity using the formula:
horizontal velocity = (g * t) / cos(θ)
Substituting the known values:
horizontal velocity = (9.8 m/s^2 * 0.782 s) / cos(θ)
Finally, we can calculate the horizontal distance traveled by multiplying the horizontal velocity by the time:
horizontal distance traveled = horizontal velocity * t
Plug in the values and calculate to find the result.