When an object falls under constant acceleration due to gravity, its average velocity can be determined based on its displacement and the time it takes to fall.
In the case of an object with zero initial velocity, falling under the influence of gravity, the average velocity can be calculated using the following formula:
Average velocity = (Final velocity + Initial velocity) / 2
Since the object starts from rest (zero initial velocity), the initial velocity is 0 m/s. The final velocity can be determined using the equation of motion:
Final velocity = Initial velocity + (acceleration * time)
In the case of free fall, the acceleration due to gravity is approximately 9.8 m/s², and the time it takes for the object to fall a certain distance can be determined using the kinematic equation:
Displacement = (Initial velocity * time) + (0.5 * acceleration * time²)
For simplicity, let's assume the object falls a distance 'd'. The displacement in this case is 'd', and the equation becomes:
d = 0.5 * acceleration * time²
Rearranging the equation to solve for time:
time = sqrt((2 * d) / acceleration)
Substituting this time value into the equation for final velocity:
Final velocity = 0 + (acceleration * time)
Since the object is falling downward, the final velocity will be negative.
Now, we can calculate the average velocity:
Average velocity = (Final velocity + Initial velocity) / 2
Average velocity = (0 + (acceleration * time)) / 2
Average velocity = (acceleration * time) / 2
Substituting the values we have:
Average velocity = (acceleration * sqrt((2 * d) / acceleration)) / 2
Simplifying the expression, we get:
Average velocity = sqrt(2 * d * acceleration)
Therefore, the average velocity of an object with zero initial velocity, falling under constant acceleration due to gravity, is given by the square root of 2 times the product of the distance fallen (displacement) and the acceleration due to gravity.