To solve this problem, we can use the equations of motion for an object in free fall.
Let's assume the positive direction for displacement is downwards (towards the floor). We can use the following equation to calculate the time of flight:
h = ut + (1/2)gt^2
Where: h = vertical displacement (4 ft) u = initial vertical velocity (unknown) g = acceleration due to gravity (-32 ft/s^2) t = time of flight (unknown)
Plugging in the values, we have:
4 = ut - (1/2)(32)t^2 4 = ut - 16t^2
Now, let's consider the horizontal motion. The horizontal displacement is given as 6 ft, and there is no acceleration horizontally. We can use the equation:
s = ut
Where: s = horizontal displacement (6 ft) u = initial horizontal velocity (unknown) t = time of flight (from the previous calculation)
Plugging in the values, we have:
6 = u * t
Now, we have a system of equations:
4 = ut - 16t^2 6 = u * t
We can solve these equations simultaneously to find the values of t (time of flight) and u (initial velocity).
From the second equation, we can express u in terms of t:
u = 6 / t
Substituting this value into the first equation:
4 = (6 / t) * t - 16t^2 4 = 6 - 16t^2 16t^2 = 2 t^2 = 2/16 t^2 = 1/8 t = 1/√8 t = √2 / 4
Substituting this value of t back into the second equation to find u:
6 = u * (√2 / 4) u = 6 * (4 / √2) u = 12 * (√2 / √2) u = 12 ft/s
So, the time of flight is approximately √2 / 4 seconds, and the initial velocity is 12 ft/s.