From a height of 2.15 m above the floor of Boston's Fleet Center, forward Paul Pierce tosses a shot straight up next to the basketball hoop with a KE of 5.40 J. If his regulation-size. basketball has a mass of 0.600 kg, will his shot go as high as the 3.04 m hoop? Use the law of conservation of energy
To determine if Paul Pierce's shot will reach the height of the 3.04 m hoop, we can use the law of conservation of energy.
The law of conservation of energy states that the total mechanical energy of a system remains constant if no external forces are acting on it. In this case, we can assume that there are no external forces acting on the basketball.
The mechanical energy of the basketball includes both kinetic energy (KE) and potential energy (PE).
Given information:
Initial height (h1) = 2.15 m (height above the floor)
Final height (h2) = 3.04 m (height of the basketball hoop)
KE = 5.40 J
Mass of the basketball (m) = 0.600 kg
The initial mechanical energy (E1) is equal to the sum of kinetic energy and potential energy at the initial position:
E1 = KE1 + PE1
Since the ball is tossed straight up, at the highest point its velocity is momentarily 0, so the kinetic energy at this point is 0. Therefore, the initial energy is equal to the potential energy at the initial height:
E1 = PE1 = m * g * h1
The final mechanical energy (E2) is equal to the sum of kinetic energy and potential energy at the final position:
E2 = KE2 + PE2
Since the ball is at the highest point, its potential energy at this point is equal to the mechanical energy at the highest point:
E2 = PE2 = m * g * h2
Using the law of conservation of energy:
E1 = E2
m * g * h1 = m * g * h2
Since the mass (m) and acceleration due to gravity (g) are constant, we can cancel them out:
h1 = h2
Comparing the heights, we can conclude that the basketball will reach the same height as the height of the basketball hoop. Therefore, Paul Pierce's shot will go as high as the 3.04 m hoop.
To determine whether Paul Pierce's shot will go as high as the 3.04 m hoop, we can use the law of conservation of energy. According to this law, the initial kinetic energy (KE) of the shot will be converted into potential energy (PE) and vice versa.
The initial kinetic energy of the shot can be calculated using the formula:
KE = (1/2) * mass * velocity^2
Since we are given the kinetic energy and the mass of the basketball, we need to find the initial velocity of the shot. Let's use this equation to solve for velocity:
5.4 J = (1/2) * 0.6 kg * velocity^2
Simplifying the equation, we have:
5.4 J = 0.3 kg * velocity^2
Dividing both sides of the equation by 0.3 kg, we get:
velocity^2 = 18 J/kg
Taking the square root of both sides, we find:
velocity = √(18 J/kg)
Now that we have the initial velocity of the shot, we can determine the maximum height it will reach using the conservation of energy.
The total mechanical energy E (sum of kinetic energy and potential energy) remains constant if no external forces are present. So, the initial kinetic energy of the shot will be equal to the potential energy at the maximum height.
KE = PE
(1/2) * mass * velocity^2 = mass * g * height
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values into the equation:
(1/2) * 0.6 kg * (√(18 J/kg))^2 = 0.6 kg * 9.8 m/s^2 * height
Simplifying, we have:
0.9 J = 5.88 m * height
Dividing both sides of the equation by 5.88 m, we get:
height = 0.9 J / 5.88 m ≈ 0.153 m
Therefore, the shot will not go as high as the 3.04 m hoop. It will reach a maximum height of approximately 0.153 m.