The distance, d, that an object falls is directly proportional to the square of time, t, it has been in free fall. An object that has been in free fall for 4 seconds has fallen 256 feet. Determine the distance the object has fallen if it has been falling for 2 seconds.
d= ___ft
d = k t^2
when d = 256, t = 4
256 = k(16)
d = 16 t^2
when t = 2
d = .....
To determine the distance the object has fallen if it has been falling for 2 seconds, we need to use the given information and the concept of direct variation.
We are given that the distance, d, an object falls is directly proportional to the square of the time, t. Mathematically, this can be represented as:
d ∝ t^2
Using direct variation, we can introduce a constant of proportionality, k, to formulate the equation:
d = k * t^2
To find the value of k, we can substitute the given values into the equation. We know that when t = 4 seconds, d = 256 feet. Plugging these values into the equation:
256 = k * 4^2
256 = k * 16
To solve for k, we divide both sides of the equation by 16:
k = 256 / 16
k = 16
Now that we have the value of k, we can use it to determine the distance, d, when t = 2 seconds. Plugging these values into the equation:
d = 16 * 2^2
d = 16 * 4
d = 64 feet
Therefore, the object will have fallen 64 feet if it has been falling for 2 seconds.