The student says that a ball dropped from a height of 3.0 meters would have less than 96 joules of kinetic energy when it hits the ground. Is she correct and why?
To determine if the student is correct, we can use the formula for gravitational potential energy:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
Since the student mentioned that the ball was dropped, we can assume that it has negligible initial velocity, thus the initial kinetic energy is 0 joules. The potential energy at the height of 3.0 meters would be:
PE = m * g * h
Given that the mass and gravity are constant, we can disregard them for this comparison. Therefore, the potential energy is directly proportional to the height:
PE ∝ h
Since kinetic energy is equal to potential energy when the object is in free fall, we can conclude that the kinetic energy is also directly proportional to the height.
KE ∝ h
Therefore, if the height is reduced, the kinetic energy will also be reduced. In this case, if the ball is dropped from a height of 3.0 meters, the kinetic energy would be less than 96 joules, as stated by the student. Hence, the student's statement is correct.
To determine if the student is correct, we need to calculate the kinetic energy of the ball when it hits the ground.
The formula for calculating kinetic energy is:
KE = (1/2) * mass * velocity^2
However, in this case, we know the height from which the ball is dropped, which allows us to calculate the velocity just before it hits the ground using the equation for gravitational potential energy:
PE = mass * g * height
Here, g is the acceleration due to gravity, which is approximately equal to 9.8 m/s^2.
By rearranging the formula for gravitational potential energy, we can solve for the velocity:
velocity = sqrt( (2 * PE) / mass )
Using the given height of 3.0 meters, we can calculate the potential energy before the ball hits the ground:
PE = mass * g * height
= mass * 9.8 m/s^2 * 3.0 m
= 29.4 * mass joules
Therefore, the velocity just before the ball hits the ground is:
velocity = sqrt( (2 * PE) / mass )
= sqrt( (2 * 29.4 * mass) / mass )
= sqrt(2 * 29.4)
= sqrt(58.8)
≈ 7.67 m/s
Now, we can calculate the kinetic energy using the formula:
KE = (1/2) * mass * velocity^2
= (1/2) * mass * (7.67 m/s)^2
= 0.5 * mass * 58.8 m^2/s^2
= 29.4 * mass joules
From the calculation, we can see that the kinetic energy depends on the mass of the ball and does not involve the height from which it is dropped. Therefore, the student is incorrect in saying that a ball dropped from a height of 3.0 meters would have less than 96 joules of kinetic energy when it hits the ground.