Ralph has the same mass of 72.6 kg, and his maximum velocity is still 7.3 m/s on the swing. What is the maximum height that Ralph reaches on his swing?
To find the maximum height that Ralph reaches on his swing, we need to use the conservation of mechanical energy. The total mechanical energy of the system remains constant, which means that the sum of Ralph's kinetic energy and potential energy remains the same at all points during his swing.
The initial mechanical energy of the system is given by the sum of Ralph's gravitational potential energy at the lowest point and his kinetic energy at that point. We can calculate Ralph's initial gravitational potential energy using the formula:
Potential Energy = mass * gravity * height
where mass is Ralph's mass, gravity is the acceleration due to gravity (approximately 9.8 m/s^2), and height is the height at the lowest point, which is zero as Ralph starts from the ground. Thus, the initial potential energy is:
Initial Potential Energy = 72.6 kg * 9.8 m/s^2 * 0 m = 0 J
Ralph's initial kinetic energy can be calculated using the formula:
Kinetic Energy = (1/2) * mass * velocity^2
where mass is Ralph's mass and velocity is his maximum velocity on the swing. So, the initial kinetic energy is:
Initial Kinetic Energy = (1/2) * 72.6 kg * (7.3 m/s)^2 = 2004.825 J
Since the total mechanical energy remains constant, the sum of the initial potential energy and initial kinetic energy is equal to the sum of the maximum potential energy and maximum kinetic energy at the highest point of the swing. At the highest point, Ralph's kinetic energy is zero because he momentarily stops moving. Thus, the maximum potential energy is equal to the total mechanical energy:
Maximum Potential Energy = Total Mechanical Energy - Initial Kinetic Energy
= 0 J - 2004.825 J
= -2004.825 J
Given that potential energy is zero at the lowest point, we can calculate the height at the highest point using the formula:
Maximum Potential Energy = mass * gravity * height
Solving for height, we have:
Height = Maximum Potential Energy / (mass * gravity)
Height = -2004.825 J / (72.6 kg * 9.8 m/s^2)
≈ -2.64 m
Since height should be positive, we can ignore the negative sign. Therefore, the maximum height Ralph reaches on his swing is approximately 2.64 meters.