Question

write a formula for a linear function that models the situation. state what the input variable represents and the domain of the function.

a stone is dropped from a water tower and its velocity increases at a rate of 32 feet per second². the stone hits the ground with a velocity of 256 feet per second.

write the formula for v(t).

v(t) = ☐

Explanation:

Step1: Identify the type of motion

The stone is in free - fall (dropped, so initial velocity \(v_0 = 0\)) and the acceleration \(a = 32\) feet per second². The velocity function for an object in free - fall with constant acceleration is given by the formula \(v(t)=v_0+at\). Since \(v_0 = 0\) (dropped from rest) and \(a = 32\) ft/s², the velocity function is a linear function of time \(t\).

Step2: Substitute values into the velocity formula

We know that for linear motion with constant acceleration, the velocity as a function of time \(t\) (where \(t\) is the time in seconds after the stone is dropped) is \(v(t)=v_0 + at\). Here, the initial velocity \(v_0 = 0\) (because the stone is dropped, not thrown) and the acceleration \(a = 32\) ft/s². Substituting these values into the formula, we get \(v(t)=0 + 32t\), which simplifies to \(v(t)=32t\). We can also verify with the final velocity. When the stone hits the ground, \(v = 256\) ft/s. Using \(v(t)=32t\), we can solve for \(t\) when \(v = 256\): \(256=32t\), so \(t = 8\) seconds. This makes sense as the time taken to reach the ground.

Answer:

\(v(t)=32t\)