Let's denote the height from which the ball falls as "h." We can use the equations of motion to solve the problem.
The first thing we need to determine is the time it takes for the ball to fall from rest to the ground. We know that the ball is in free fall, meaning it is only influenced by gravity. The equation that relates distance, time, and acceleration for an object in free fall is:
Distance = (1/2) × Acceleration × Time²
In this case, the distance is "h," the acceleration is the acceleration due to gravity (approximately 9.8 m/s²), and we want to solve for time.
h = (1/2) × 9.8 × Time²
Simplifying the equation:
h = 4.9 × Time²
Next, we are given that the ball travels half its path in the last second of its free fall. Let's denote the time it takes for the ball to travel half its path as "t."
Therefore, we have:
h/2 = 4.9 × t²
Now, let's consider the last second of free fall. During this time, the ball covers the remaining half of the total distance. So, the time taken for the last second is also "t."
We can set up the equation for the last second:
h/2 = 4.9 × t²
From this equation, we can solve for "h":
h = 2 × 4.9 × t²
Since the ball travels half its path during the last second, the total time of free fall is twice the time taken for the last second. So, we can express "t" in terms of the total time of free fall, which we'll call "T":
t = T/2
Substituting this value into the equation for "h":
h = 2 × 4.9 × (T/2)²
Simplifying:
h = 4.9 × (T/2)²
h = 4.9 × T²/4
h = 1.225 × T²
Therefore, the height from which the ball falls is given by 1.225 times the square of the total time of free fall, "T."