the largest watermelon ever grown had a mass of 118 kg. suppose this watermelon were exhibited on a platform 5 m above the ground . after the exhibition, the watermelon is allowed to slide along to the ground along a smooth ramp, How high above the ground is the watermelon at the moment its kinetic energy is 4.61 J ?
To determine the height above the ground when the watermelon's kinetic energy is 4.61 J, we can use the principle of conservation of energy.
The total mechanical energy of the watermelon is conserved throughout its motion. This includes both its potential energy (due to its position) and its kinetic energy (due to its motion). The equation for conservation of energy in this system is:
Potential Energy + Kinetic Energy = Total Mechanical Energy
Initially, when the watermelon is on the platform before sliding, it has potential energy due to its height above the ground. The equation for potential energy is:
Potential Energy = mass * gravitational acceleration * height
Given that the mass of the watermelon is 118 kg and the gravitational acceleration is approximately 9.8 m/s^2, we can calculate the initial potential energy:
Potential Energy = 118 kg * 9.8 m/s^2 * 5 m = 5686 J
Since the watermelon slides down the ramp along a smooth path, all its initial potential energy will be converted to kinetic energy at the bottom of the ramp. So, the equation becomes:
Kinetic Energy = Total Mechanical Energy - Potential Energy
Given that the kinetic energy is 4.61 J and the initial potential energy is 5686 J, we can rearrange the equation to find the total mechanical energy:
Total Mechanical Energy = Kinetic Energy + Potential Energy
= 4.61 J + 5686 J
= 5689.61 J
Now, we can use the total mechanical energy to find the final potential energy when the watermelon reaches the height above the ground at 4.61 J of kinetic energy:
Potential Energy = Total Mechanical Energy - Kinetic Energy
= 5689.61 J - 4.61 J
= 5685 J
Finally, we can plug in the values of mass, gravitational acceleration, and the calculated potential energy into the equation for potential energy to find the height above the ground:
Potential Energy = mass * gravitational acceleration * height
5685 J = 118 kg * 9.8 m/s^2 * height
Solving for height:
height = 5685 J / (118 kg * 9.8 m/s^2)
Calculating the height:
height ≈ 4.89 m
Therefore, at the moment when the watermelon's kinetic energy is 4.61 J, it is approximately 4.89 meters above the ground.
To determine the height above the ground when the watermelon's kinetic energy is 4.61 J, we need to apply the principle of conservation of mechanical energy.
The potential energy at the top of the ramp is given by:
PE_top = m * g * h
Where:
m = mass of the watermelon = 118 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height above the ground
The initial potential energy can be determined by multiplying the mass of the watermelon by the gravitational acceleration and the height above the ground on the platform:
PE_initial = m * g * 5 m
The final kinetic energy is given as 4.61 J.
Since the mechanical energy is conserved, we can equate the initial potential energy to the final kinetic energy:
PE_initial = KE_final
Therefore:
m * g * 5 m = 4.61 J
Solving for h:
h = 4.61 J / (m * g)
Substituting the given values:
h = 4.61 J / (118 kg * 9.8 m/s^2)
Calculating h:
h ≈ 0.0040 m
Therefore, the watermelon is approximately 0.0040 m (or 4.0 mm) above the ground when its kinetic energy is 4.61 J.
If its kinetic energy is 4.61 J, its potential energy will have fallen by the same amount (neglecting friction and assuming it does not roll).
Amount of drop = 4.61/(M*g) = 3.99*10^-3 m
(about 4.0 millimeters)
Height above ground = 4.996 m