Question

Use a vertical motion model to find how long it will take for the object to reach the ground. You throw a ball downward with an initial speed of 10 feet per second out of a window to a friend 20 feet below. Your friend does not catch the ball.

Short answer

It will take approximately 1.264 seconds for the ball to reach the ground.

Step-by-step solution

Write down the given values

Given: Initial height (\(h_0\)) is 20 feet below, thus is -20 feet. Initial velocity (\(v\)) is 10 feet/second downward, thus -10 feet/second. The acceleration due to gravity (\(g\)) is 32.2 feet/second^2. Final height (\(h\)) is 0 (ground level). We need to calculate the time (\(t\)).

Plug in the values into the equation

Substitute the given values into the equation: 0 = -20 - 10t + 0.5*(32.2)t^2. This simplifies to: 0 = -20 -10t +16.1t^2. Which can be rearranged to: 16.1t^2 -10t - 20 = 0.

Apply the Quadratic Formula

Solve the quadratic equation with the Quadratic Formula: \(t = (-b±\sqrt{b^2 - 4ac}) / (2a)\), which gives: \(t = (10±\sqrt{(-10)^2 - 4*16.1*(-20)}) / (2*16.1)\). However, since \(t\) cannot be negative we discard the negative solution, resulting in: \( t = (10 + \sqrt{100 + 1296}) / 32.2 \) .

Calculate the Time

Now, calculate the value for \(t\): \( t = (10 + \sqrt{1396}) / 32.2 \). This simplifies further to: \(t = 1.264\) seconds when rounded to three decimal places.