Question

In Exercises \(64-66,\) use the vertical motion model \(\boldsymbol{h}=-\mathbf{1 6 t}^{2}+\boldsymbol{v t}+\boldsymbol{s},\) where \(\boldsymbol{h}\) is the height (in feet), \(t\) is the time in motion (in seconds), \(v\) is the initial velocity (in feet per second), and \(s\) is the initial height (in feet). Solve by factoring. A gymnast dismounts the uneven parallel bars at a height of 8 feet with an initial upward velocity of 8 feet per second. Find the time \(t\) (in seconds) it takes for the gymnast to reach the ground. Is your answer reasonable?

Short answer

The gymnast will reach the ground after 0.5 seconds.

Step-by-step solution

Write down the equation

The first step is to write down the equation for the height \(h\), using the provided model: \(h=-16t^2+vt+s\). Here, the initial velocity \(v\) is 8 feet per second and the initial height \(s\) is 8 feet. Substituting these values, we get: \(h=-16t^2+8t+8\). As we are interested in the time when the gymnast hits the ground, we set \(h\) equal to 0: \(0=-16t^2+8t+8\).

Simplify the equation

The next step is to simplify the equation by dividing every term by -8: \(-2t^2 + t + 1 = 0\). This equation is easier to handle. We multiply every term by -1 to make the first term positive, and we end up with \(2t^2 - t - 1 = 0\).

Factorise the equation

Now, we factorize the equation. This is a quadratic equation in the form \(at^2+bt+c=0\), and factorizes to \((2t - 1)(t+1) = 0\).

Find out the solutions

We set each factor equal to zero and solve for \(t\). This gives us \(t = 1/2\) from \(2t - 1 = 0\) and \(t = -1\) from \(t+1=0\).

Validate the solutions

Negative time doesn't have any physical sense, so it must be rejected. Therefore, the gymnast will reach the ground after \(t = 1/2\) second.