To find the height at which the kinetic energy of the ball becomes half, we need to use the principle of conservation of energy. The total mechanical energy of the ball is conserved throughout its motion.
The mechanical energy of the ball consists of its kinetic energy (KE) and potential energy (PE) due to its height. At any point during its motion, the total mechanical energy remains constant.
Initially, the ball has only kinetic energy as it is moving vertically upwards. Let's denote the initial kinetic energy as KE_i.
At the highest point of the ball's motion, its velocity becomes zero. Therefore, all the initial kinetic energy is converted into potential energy. The height at this point is the maximum height reached by the ball.
At this maximum height, the potential energy (PE_max) is equal to the initial kinetic energy (KE_i).
Now, we want to find the height at which the kinetic energy becomes half of the initial kinetic energy (KE_i/2).
At any height h, the potential energy is given by the formula:
PE = m * g * h
where m is the mass of the ball and g is the acceleration due to gravity.
Since the total mechanical energy is conserved, we can equate the potential energy at the maximum height (PE_max) to the potential energy at the desired height (PE):
PE_max = PE
m * g * h_max = m * g * h
Canceling out the mass (m) and acceleration due to gravity (g) on both sides:
h_max = h
So, the height at which the kinetic energy becomes half is equal to the maximum height reached by the ball.
To calculate the maximum height, we can use the equation for potential energy:
PE_max = KE_i
m * g * h_max = KE_i
h_max = KE_i / (m * g)
Given that the initial velocity (v_i) is 20 m/s, we can calculate the initial kinetic energy (KE_i) using the formula:
KE_i = (1/2) * m * v_i^2
Substituting the values, we get:
KE_i = (1/2) * m * (20)^2
Since the mass (m) cancels out when calculating the height, we can disregard it.
Finally, substituting the values into the equation for the maximum height:
h_max = KE_i / (m * g)
h_max = [(1/2) * m * (20)^2] / (m * g)
h_max = (1/2) * (20)^2 / g
Now, we need to know the value of the acceleration due to gravity (g) to calculate the maximum height. The standard value for acceleration due to gravity is approximately 9.8 m/s^2.
Let's calculate the maximum height using this value:
h_max = (1/2) * (20)^2 / 9.8
h_max = 200 / 9.8
h_max ≈ 20.41 meters
Therefore, the height at which the kinetic energy becomes half is approximately 20.41 meters.